diff -r f04b33ce250f -r a4dc62a46ee4 ex/cla.ML --- a/ex/cla.ML Tue Oct 24 14:59:17 1995 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,455 +0,0 @@ -(* Title: HOL/ex/cla - ID: $Id$ - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1994 University of Cambridge - -Higher-Order Logic: predicate calculus problems - -Taken from FOL/cla.ML; beware of precedence of = vs <-> -*) - -writeln"File HOL/ex/cla."; - -goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)"; -by (fast_tac HOL_cs 1); -result(); - -(*If and only if*) - -goal HOL.thy "(P=Q) = (Q=P::bool)"; -by (fast_tac HOL_cs 1); -result(); - -goal HOL.thy "~ (P = (~P))"; -by (fast_tac HOL_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 HOL.thy "(P-->Q) = (~Q --> ~P)"; -by (fast_tac HOL_cs 1); -result(); - -(*2*) -goal HOL.thy "(~ ~ P) = P"; -by (fast_tac HOL_cs 1); -result(); - -(*3*) -goal HOL.thy "~(P-->Q) --> (Q-->P)"; -by (fast_tac HOL_cs 1); -result(); - -(*4*) -goal HOL.thy "(~P-->Q) = (~Q --> P)"; -by (fast_tac HOL_cs 1); -result(); - -(*5*) -goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))"; -by (fast_tac HOL_cs 1); -result(); - -(*6*) -goal HOL.thy "P | ~ P"; -by (fast_tac HOL_cs 1); -result(); - -(*7*) -goal HOL.thy "P | ~ ~ ~ P"; -by (fast_tac HOL_cs 1); -result(); - -(*8. Peirce's law*) -goal HOL.thy "((P-->Q) --> P) --> P"; -by (fast_tac HOL_cs 1); -result(); - -(*9*) -goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; -by (fast_tac HOL_cs 1); -result(); - -(*10*) -goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"; -by (fast_tac HOL_cs 1); -result(); - -(*11. Proved in each direction (incorrectly, says Pelletier!!) *) -goal HOL.thy "P=P::bool"; -by (fast_tac HOL_cs 1); -result(); - -(*12. "Dijkstra's law"*) -goal HOL.thy "((P = Q) = R) = (P = (Q = R))"; -by (fast_tac HOL_cs 1); -result(); - -(*13. Distributive law*) -goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))"; -by (fast_tac HOL_cs 1); -result(); - -(*14*) -goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))"; -by (fast_tac HOL_cs 1); -result(); - -(*15*) -goal HOL.thy "(P --> Q) = (~P | Q)"; -by (fast_tac HOL_cs 1); -result(); - -(*16*) -goal HOL.thy "(P-->Q) | (Q-->P)"; -by (fast_tac HOL_cs 1); -result(); - -(*17*) -goal HOL.thy "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Classical Logic: examples with quantifiers"; - -goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))"; -by (fast_tac HOL_cs 1); -result(); - -goal HOL.thy "(? x. P-->Q(x)) = (P --> (? x.Q(x)))"; -by (fast_tac HOL_cs 1); -result(); - -goal HOL.thy "(? x.P(x)-->Q) = ((! x.P(x)) --> Q)"; -by (fast_tac HOL_cs 1); -result(); - -goal HOL.thy "((! x.P(x)) | Q) = (! x. P(x) | Q)"; -by (fast_tac HOL_cs 1); -result(); - -(*From Wishnu Prasetya*) -goal HOL.thy - "(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \ -\ --> p(t) | r(t)"; -by (fast_tac HOL_cs 1); -result(); - - -writeln"Problems requiring quantifier duplication"; - -(*Needs multiple instantiation of the quantifier.*) -goal HOL.thy "(! x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; -by (deepen_tac HOL_cs 1 1); -result(); - -(*Needs double instantiation of the quantifier*) -goal HOL.thy "? x. P(x) --> P(a) & P(b)"; -by (deepen_tac HOL_cs 1 1); -result(); - -goal HOL.thy "? z. P(z) --> (! x. P(x))"; -by (deepen_tac HOL_cs 1 1); -result(); - -goal HOL.thy "? x. (? y. P(y)) --> P(x)"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Hard examples with quantifiers"; - -writeln"Problem 18"; -goal HOL.thy "? y. ! x. P(y)-->P(x)"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 19"; -goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 20"; -goal HOL.thy "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w))) \ -\ --> (? x y. P(x) & Q(y)) --> (? z. R(z))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 21"; -goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 22"; -goal HOL.thy "(! x. P = Q(x)) --> (P = (! x. Q(x)))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 23"; -goal HOL.thy "(! x. P | Q(x)) = (P | (! x. Q(x)))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 24"; -goal HOL.thy "~(? 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 (fast_tac HOL_cs 1); -result(); - -writeln"Problem 25"; -goal HOL.thy "(? 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))) \ -\ --> (? x. Q(x)&P(x))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 26"; -goal HOL.thy "((? 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 (fast_tac HOL_cs 1); -result(); - -writeln"Problem 27"; -goal HOL.thy "(? 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))) \ -\ --> (! x. M(x) --> ~L(x))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 28. AMENDED"; -goal HOL.thy "(! 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))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; -goal HOL.thy "(? 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 (fast_tac HOL_cs 1); -result(); - -writeln"Problem 30"; -goal HOL.thy "(! x. P(x) | Q(x) --> ~ R(x)) & \ -\ (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ -\ --> (! x. S(x))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 31"; -goal HOL.thy "~(? x.P(x) & (Q(x) | R(x))) & \ -\ (? x. L(x) & P(x)) & \ -\ (! x. ~ R(x) --> M(x)) \ -\ --> (? x. L(x) & M(x))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 32"; -goal HOL.thy "(! 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))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 33"; -goal HOL.thy "(! x. P(a) & (P(x)-->P(b))-->P(c)) = \ -\ (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 34 AMENDED (TWICE!!) NOT PROVED AUTOMATICALLY"; -(*Andrews's challenge*) -goal HOL.thy "((? x. ! y. p(x) = p(y)) = \ -\ ((? x. q(x)) = (! y. p(y)))) = \ -\ ((? x. ! y. q(x) = q(y)) = \ -\ ((? x. p(x)) = (! y. q(y))))"; -by (deepen_tac HOL_cs 3 1); -(*slower with smaller bounds*) -result(); - -writeln"Problem 35"; -goal HOL.thy "? x y. P(x,y) --> (! u v. P(u,v))"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 36"; -goal HOL.thy "(! 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))) \ -\ --> (! x. ? y. H(x,y))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 37"; -goal HOL.thy "(! 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))) \ -\ --> (! x. ? y. R(x,y))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 38"; -goal HOL.thy - "(! 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))) & \ -\ (~p(a) | ~(? y. p(y) & r(x,y)) | \ -\ (? z. ? w. p(z) & r(x,w) & r(w,z))))"; - -writeln"Problem 39"; -goal HOL.thy "~ (? x. ! y. F(y,x) = (~F(y,y)))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 40. AMENDED"; -goal HOL.thy "(? y. ! x. F(x,y) = F(x,x)) \ -\ --> ~ (! x. ? y. ! z. F(z,y) = (~F(z,x)))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 41"; -goal HOL.thy "(! z. ? y. ! x. f(x,y) = (f(x,z) & ~ f(x,x))) \ -\ --> ~ (? z. ! x. f(x,z))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 42"; -goal HOL.thy "~ (? y. ! x. p(x,y) = (~ (? z. p(x,z) & p(z,x))))"; -by (deepen_tac HOL_cs 3 1); -result(); - -writeln"Problem 43 NOT PROVED AUTOMATICALLY"; -goal HOL.thy - "(! x::'a. ! y::'a. q(x,y) = (! z. p(z,x) = (p(z,y)::bool))) \ -\ --> (! x. (! y. q(x,y) = (q(y,x)::bool)))"; - - -writeln"Problem 44"; -goal HOL.thy "(! 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))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 45"; -goal HOL.thy - "(! 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)) & \ -\ (? x. f(x) & (! y. h(x,y) --> l(y)) \ -\ & (! y. g(y) & h(x,y) --> j(x,y))) \ -\ --> (? x. f(x) & ~ (? y. g(y) & h(x,y)))"; -by (best_tac HOL_cs 1); -result(); - - -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"; -by (fast_tac HOL_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 HOL.thy "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \ -\ --> (! u::'a.P(u))"; -by (Classical.safe_tac HOL_cs); -by (res_inst_tac [("x","a")] allE 1); -by (assume_tac 1); -by (res_inst_tac [("x","b")] allE 1); -by (assume_tac 1); -by (fast_tac HOL_cs 1); -result(); - -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))"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 51"; -goal HOL.thy - "(? z w. ! x y. P(x,y) = (x=z & y=w)) --> \ -\ (? z. ! x. ? w. (! y. P(x,y) = (y=w)) = (x=z))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 52"; -(*Almost the same as 51. *) -goal HOL.thy - "(? z w. ! x y. P(x,y) = (x=z & y=w)) --> \ -\ (? w. ! y. ? z. (! x. P(x,y) = (x=z)) = (y=w))"; -by (best_tac HOL_cs 1); -result(); - -writeln"Problem 55"; - -(*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) & \ -\ (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)) & \ -\ (hates(agatha,agatha) & hates(agatha,charles)) & \ -\ (!x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \ -\ (!x. hates(agatha,x) --> hates(butler,x)) & \ -\ (!x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \ -\ killed(?who,agatha)"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 56"; -goal HOL.thy - "(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Problem 57"; -goal HOL.thy - "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 (fast_tac HOL_cs 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)))"; -val f_cong = read_instantiate [("f","f")] arg_cong; -by (fast_tac (HOL_cs addIs [f_cong]) 1); -result(); - -writeln"Problem 59"; -goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))"; -by (deepen_tac HOL_cs 1 1); -result(); - -writeln"Problem 60"; -goal HOL.thy - "! x. P(x,f(x)) = (? y. (! z. P(z,y) --> P(z,f(x))) & P(x,y))"; -by (fast_tac HOL_cs 1); -result(); - -writeln"Reached end of file.";