src/HOL/ex/mesontest.ML

Changed argument order of nat_rec.

(*  Title:      HOL/ex/mesontest
    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;

keep_derivs := MinDeriv;
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 goes = gocls clauses;

goal HOL.thy "False";
by (resolve_tac goes 1);
keep_derivs := FullDeriv;

by (prolog_step_tac horns 1);
by (iter_deepen_prolog_tac horns);
by (depth_prolog_tac horns);
by (best_prolog_tac size_of_subgoals horns);
*)

writeln"File HOL/ex/meson-test.";

(*Deep unifications can be required, esp. during transformation to clauses*)
val orig_trace_bound = !Unify.trace_bound
and orig_search_bound = !Unify.search_bound;
Unify.trace_bound := 20;
Unify.search_bound := 40;

(**** 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::_ = gocls clauses25;

goal HOL.thy "False";
by (rtac 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::_ = gocls clauses26;

goal HOL.thy "False";
by (rtac go26 1);
by (depth_prolog_tac horns26);  (*1.4 secs*)
(*Proof is of length 107!!*)


writeln"Problem 43  NOW PROVED AUTOMATICALLY!!";  (*16 Horn clauses*)
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::_ = gocls clauses43;

goal HOL.thy "False";
by (rtac go43 1);
by (best_prolog_tac size_of_subgoals horns43);   (*1.6 secs*)

(* 
#1  (q x xa ==> ~ q x xa) ==> q xa x
#2  (q xa x ==> ~ q xa x) ==> q x xa
#3  (~ q x xa ==> q x xa) ==> ~ q xa x
#4  (~ q xa x ==> q xa x) ==> ~ q x xa
#5  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?U ==> p ?W ?U |] ==> p ?W ?V
#6  [| ~ p ?W ?U ==> p ?W ?U; p ?W ?V ==> ~ p ?W ?V |] ==> ~ q ?U ?V
#7  [| p ?W ?V ==> ~ p ?W ?V; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?U
#8  [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?V ==> p ?W ?V |] ==> p ?W ?U
#9  [| ~ p ?W ?V ==> p ?W ?V; p ?W ?U ==> ~ p ?W ?U |] ==> ~ q ?U ?V
#10 [| p ?W ?U ==> ~ p ?W ?U; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?V
#11 [| p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U;
       p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V |] ==> q ?U ?V
#12 [| p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
    p (xb ?U ?V) ?U
#13 [| q ?U ?V ==> ~ q ?U ?V; p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U |] ==>
    p (xb ?U ?V) ?V
#14 [| ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U;
       ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V |] ==> q ?U ?V
#15 [| ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==>
    ~ p (xb ?U ?V) ?U
#16 [| q ?U ?V ==> ~ q ?U ?V; ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U |] ==>
    ~ p (xb ?U ?V) ?V

And here is the proof!  (Unkn is the start state after use of goal clause)
[Unkn, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
   Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
   Asm 1, Res ([Thm "#13"], false, 1), Asm 1, Res ([Thm "#10"], false, 1),
   Res ([Thm "#16"], false, 1), Asm 2, Asm 1, Res ([Thm "#1"], false, 1),
   Asm 1, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1),
   Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2,
   Asm 1, Res ([Thm "#8"], false, 1), Res ([Thm "#2"], false, 1), Asm 1,
   Res ([Thm "#12"], false, 1), Asm 2, Asm 1] : lderiv list
*)


(*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 (safe_meson_tac 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(); 

goal HOL.thy "(! x. P x --> P(f x))  &  P d --> P(f(f(f d)))";
by (safe_meson_tac 1);
result();

(*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";  (*The first goal clause is useless*)
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";  (*24 Horn clauses*)
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";	(*13 Horn clauses*)
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";	(*14 Horn clauses*)
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";
	(*62 Horn clauses*)
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);          (*0.7 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";  (*10 Horn clauses; first negative clauses is useless*)
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";  (*55 Horn clauses*)
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!!)"; (*924 Horn clauses*)
(*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);          (*13 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";  (*15 Horn clauses*)
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";  (*10 Horn clauses*)
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";  (*Quite hard: 422 Horn clauses!!*)
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_best_meson_tac 1);  (*10 secs; iter. deepening is slightly slower*)
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_best_meson_tac 1);	
(*1.6 secs; iter. deepening is slightly slower*)
result();

writeln"Problem 44";  (*13 Horn clauses; 7-step proof*)
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";  (*27 Horn clauses; 54-step proof*)
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_best_meson_tac 1);	
(*1.6 secs; iter. deepening is slightly slower*)
result();

writeln"Problem 46";  (*26 Horn clauses; 21-step proof*)
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_best_meson_tac 1);	
(*1.7 secs; iter. deepening is slightly slower*)
result();

writeln"Problem 47.  Schubert's Steamroller";
	(*26 clauses; 63 Horn clauses*)
goal HOL.thy
    "(! x. P1 x --> P0 x) & (? x.P1 x) &     \
\    (! x. P2 x --> P0 x) & (? x.P2 x) &     \
\    (! x. P3 x --> P0 x) & (? x.P3 x) &     \
\    (! x. P4 x --> P0 x) & (? x.P4 x) &     \
\    (! x. P5 x --> P0 x) & (? x.P5 x) &     \
\    (! x. Q1 x --> Q0 x) & (? x.Q1 x) &     \
\    (! x. P0 x --> ((! y.Q0 y-->R x y) |    \
\                     (! y.P0 y & S y x &     \
\                          (? z.Q0 z&R y z) --> R x y))) &   \
\    (! x y. P3 y & (P5 x|P4 x) --> S x y) &        \
\    (! x y. P3 x & P2 y --> S x y) &        \
\    (! x y. P2 x & P1 y --> S x y) &        \
\    (! x y. P1 x & (P2 y|Q1 y) --> ~R x y) &       \
\    (! x y. P3 x & P4 y --> R x y) &        \
\    (! x y. P3 x & P5 y --> ~R x y) &       \
\    (! x. (P4 x|P5 x) --> (? y.Q0 y & R x y))      \
\    --> (? x y. P0 x & P0 y & (? z. Q1 z & R y z & R x y))";
by (safe_meson_tac 1);	 (*119 secs*)
result();

(*The Los problem?  Circulated by John Harrison*)
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. P x y --> P y x) &       \
\      (! (x::'a) y. P x y | Q x y)     \
\      --> (! x y. P x y) | (! x y. Q x y)";
by (safe_best_meson_tac 1);	(*3 secs; iter. deepening is VERY slow*)
result();

(*A similar 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_best_meson_tac 1);          (*2.7 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"Problem 62 as corrected in AAR newletter #31";
goal HOL.thy
    "(ALL x. p a & (p x --> p(f x)) --> p(f(f x)))  =   \
\    (ALL x. (~ p a | p x | p(f(f x))) &                        \
\            (~ p a | ~ p(f x) | p(f(f x))))";
by (safe_meson_tac 1);
result();


(** Charles Morgan's problems **)

val axa = "! x y.   T(i x(i y x))";
val axb = "! x y z. T(i(i x(i y z))(i(i x y)(i x z)))";
val axc = "! x y.   T(i(i(n x)(n y))(i y x))";
val axd = "! x y.   T(i x y) & T x --> T y";

fun axjoin ([],   q) = q
  | axjoin (p::ps, q) = "(" ^ p ^ ") --> (" ^ axjoin(ps,q) ^ ")";

goal HOL.thy (axjoin([axa,axb,axd], "! x. T(i x x)"));
by (safe_meson_tac 1);  
result();

writeln"Problem 66";  
goal HOL.thy (axjoin([axa,axb,axc,axd], "! x. T(i x(n(n x)))"));
(*TOO SLOW: more than 24 minutes!
by (safe_meson_tac 1);
result();
*)

writeln"Problem 67";  
goal HOL.thy (axjoin([axa,axb,axc,axd], "! x. T(i(n(n x)) x)"));
(*TOO SLOW: more than 3 minutes!
by (safe_meson_tac 1);
result();
*)


(*Restore original values*)
Unify.trace_bound := orig_trace_bound;
Unify.search_bound := orig_search_bound;

writeln"Reached end of file.";

(*26 August 1992: loaded in 277 secs.  New Jersey v 75*)