diff -r 000000000000 -r 7949f97df77a ex/meson-test.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ex/meson-test.ML Thu Sep 16 12:21:07 1993 +0200 @@ -0,0 +1,496 @@ +(* 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*)