(* Title: HOL/ex/meson
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Test data for the MESON proof procedure
(Excludes the equality problems 51, 52, 56, 58)
show_hyps:=false;
by (rtac ccontr 1);
val [prem] = gethyps 1;
val nnf = make_nnf prem;
val xsko = skolemize nnf;
by (cut_facts_tac [xsko] 1 THEN REPEAT (etac exE 1));
val [_,sko] = gethyps 1;
val clauses = make_clauses [sko];
val horns = make_horns clauses;
val go::_ = neg_clauses clauses;
goal HOL.thy "False";
by (rtac (make_goal go) 1);
by (prolog_step_tac horns 1);
by (depth_prolog_tac horns);
by (best_prolog_tac size_of_subgoals horns);
*)
writeln"File HOL/ex/meson-test.";
(**** Interactive examples ****)
(*Generate nice names for Skolem functions*)
Logic.auto_rename := true; Logic.set_rename_prefix "a";
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 (rtac ccontr 1);
val [prem25] = gethyps 1;
val nnf25 = make_nnf prem25;
val xsko25 = skolemize nnf25;
by (cut_facts_tac [xsko25] 1 THEN REPEAT (etac exE 1));
val [_,sko25] = gethyps 1;
val clauses25 = make_clauses [sko25]; (*7 clauses*)
val horns25 = make_horns clauses25; (*16 Horn clauses*)
val go25::_ = neg_clauses clauses25;
goal HOL.thy "False";
by (rtac (make_goal go25) 1);
by (depth_prolog_tac horns25);
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 (rtac ccontr 1);
val [prem26] = gethyps 1;
val nnf26 = make_nnf prem26;
val xsko26 = skolemize nnf26;
by (cut_facts_tac [xsko26] 1 THEN REPEAT (etac exE 1));
val [_,sko26] = gethyps 1;
val clauses26 = make_clauses [sko26]; (*9 clauses*)
val horns26 = make_horns clauses26; (*24 Horn clauses*)
val go26::_ = neg_clauses clauses26;
goal HOL.thy "False";
by (rtac (make_goal go26) 1);
by (depth_prolog_tac horns26); (*6 secs*)
writeln"Problem 43 NOW PROVED AUTOMATICALLY!!";
goal HOL.thy "(! x. ! y. q(x,y) = (! z. p(z,x) = p(z,y)::bool)) \
\ --> (! x. (! y. q(x,y) = q(y,x)::bool))";
by (rtac ccontr 1);
val [prem43] = gethyps 1;
val nnf43 = make_nnf prem43;
val xsko43 = skolemize nnf43;
by (cut_facts_tac [xsko43] 1 THEN REPEAT (etac exE 1));
val [_,sko43] = gethyps 1;
val clauses43 = make_clauses [sko43]; (*6*)
val horns43 = make_horns clauses43; (*16*)
val go43::_ = neg_clauses clauses43;
goal HOL.thy "False";
by (rtac (make_goal go43) 1);
by (best_prolog_tac size_of_subgoals horns43);
(*8.7 secs*)
(*Restore variable name preservation*)
Logic.auto_rename := false;
(**** Batch test data ****)
(*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 (safe_meson_tac 1);
result();
(*2*)
goal HOL.thy "(~ ~ P) = P";
by (safe_meson_tac 1);
result();
(*3*)
goal HOL.thy "~(P-->Q) --> (Q-->P)";
by (safe_meson_tac 1);
result();
(*4*)
goal HOL.thy "(~P-->Q) = (~Q --> P)";
by (safe_meson_tac 1);
result();
(*5*)
goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
by (safe_meson_tac 1);
result();
(*6*)
goal HOL.thy "P | ~ P";
by (safe_meson_tac 1);
result();
(*7*)
goal HOL.thy "P | ~ ~ ~ P";
by (safe_meson_tac 1);
result();
(*8. Peirce's law*)
goal HOL.thy "((P-->Q) --> P) --> P";
by (safe_meson_tac 1);
result();
(*9*)
goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
by (safe_meson_tac 1);
result();
(*10*)
goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
by (safe_meson_tac 1);
result();
(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
goal HOL.thy "P=P::bool";
by (safe_meson_tac 1);
result();
(*12. "Dijkstra's law"*)
goal HOL.thy "((P = Q) = R) = (P = (Q = R))";
by (best_meson_tac size_of_subgoals 1);
result();
(*13. Distributive law*)
goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";
by (safe_meson_tac 1);
result();
(*14*)
goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";
by (safe_meson_tac 1);
result();
(*15*)
goal HOL.thy "(P --> Q) = (~P | Q)";
by (safe_meson_tac 1);
result();
(*16*)
goal HOL.thy "(P-->Q) | (Q-->P)";
by (safe_meson_tac 1);
result();
(*17*)
goal HOL.thy "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))";
by (safe_meson_tac 1);
result();
writeln"Classical Logic: examples with quantifiers";
goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
by (safe_meson_tac 1);
result();
goal HOL.thy "(? x. P-->Q(x)) = (P --> (? x.Q(x)))";
by (safe_meson_tac 1);
result();
goal HOL.thy "(? x.P(x)-->Q) = ((! x.P(x)) --> Q)";
by (safe_meson_tac 1);
result();
goal HOL.thy "((! x.P(x)) | Q) = (! x. P(x) | Q)";
by (safe_meson_tac 1);
result();
writeln"Testing the complete tactic";
(*Not provable by pc_tac: needs multiple instantiation of !.
Could be proved trivially by a PROLOG interpreter*)
goal HOL.thy "(! x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
by (safe_meson_tac 1);
result();
(*Not provable by pc_tac: needs double instantiation of EXISTS*)
goal HOL.thy "? x. P(x) --> P(a) & P(b)";
by (safe_meson_tac 1);
result();
goal HOL.thy "? z. P(z) --> (! x. P(x))";
by (safe_meson_tac 1);
result();
writeln"Hard examples with quantifiers";
writeln"Problem 18";
goal HOL.thy "? y. ! x. P(y)-->P(x)";
by (safe_meson_tac 1);
result();
writeln"Problem 19";
goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
by (safe_meson_tac 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 (safe_meson_tac 1);
result();
writeln"Problem 21";
goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
by (safe_meson_tac 1);
result();
writeln"Problem 22";
goal HOL.thy "(! x. P = Q(x)) --> (P = (! x. Q(x)))";
by (safe_meson_tac 1);
result();
writeln"Problem 23";
goal HOL.thy "(! x. P | Q(x)) = (P | (! x. Q(x)))";
by (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 1); (*5 secs*)
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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 1); (*5.6 secs*)
result();
writeln"Problem 34 AMENDED (TWICE!!)";
(*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 (safe_meson_tac 1); (*90 secs*)
result();
writeln"Problem 35";
goal HOL.thy "? x y. P(x,y) --> (! u v. P(u,v))";
by (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 1); (*causes unification tracing messages*)
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))))";
by (safe_meson_tac 1); (*62 secs*)
result();
writeln"Problem 39";
goal HOL.thy "~ (? x. ! y. F(y,x) = (~F(y,y)))";
by (safe_meson_tac 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 (safe_meson_tac 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 (safe_meson_tac 1);
result();
writeln"Problem 42";
goal HOL.thy "~ (? y. ! x. p(x,y) = (~ (? z. p(x,z) & p(z,x))))";
by (safe_meson_tac 1);
result();
writeln"Problem 43 NOW PROVED AUTOMATICALLY!!";
goal HOL.thy "(! x. ! y. q(x,y) = (! z. p(z,x) = p(z,y)::bool)) \
\ --> (! x. (! y. q(x,y) = q(y,x)::bool))";
by (safe_meson_tac 1);
result();
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 (safe_meson_tac 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 (safe_meson_tac 1); (*11 secs*)
result();
writeln"Problem 46";
goal HOL.thy
"(! x. f(x) & (! y. f(y) & h(y,x) --> g(y)) --> g(x)) & \
\ ((? x.f(x) & ~g(x)) --> \
\ (? x. f(x) & ~g(x) & (! y. f(y) & ~g(y) --> j(x,y)))) & \
\ (! x y. f(x) & f(y) & h(x,y) --> ~j(y,x)) \
\ --> (! x. f(x) --> g(x))";
by (safe_meson_tac 1); (*11 secs*)
result();
(* Example suggested by Johannes Schumann and credited to Pelletier *)
goal HOL.thy "(!x y z. P(x,y) --> P(y,z) --> P(x,z)) --> \
\ (!x y z. Q(x,y) --> Q(y,z) --> Q(x,z)) --> \
\ (!x y.Q(x,y) --> Q(y,x)) --> (!x y. P(x,y) | Q(x,y)) --> \
\ (!x y.P(x,y)) | (!x y.Q(x,y))";
by (safe_meson_tac 1); (*32 secs*)
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 (safe_meson_tac 1);
result();
writeln"Problem 55";
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
meson_tac cannot report 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)) --> \
\ (? x. killed(x,agatha))";
by (safe_meson_tac 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 (safe_meson_tac 1);
result();
writeln"Problem 58";
(* Challenge found on info-hol *)
goal HOL.thy
"! P Q R x. ? v w. ! y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))";
by (safe_meson_tac 1);
result();
writeln"Problem 59";
goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
by (safe_meson_tac 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 (safe_meson_tac 1);
result();
writeln"Reached end of file.";
(*26 August 1992: loaded in 277 secs. New Jersey v 75*)