src/HOL/ex/mesontest.ML

 (*Generate nice names for Skolem functions*)
 Logic.auto_rename := true; Logic.set_rename_prefix "a";
 
 
 writeln"Problem 25";
-Goal "(? 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)";
+Goal "(\\<exists>x. P x) &  \
+\     (\\<forall>x. L x --> ~ (M x & R x)) &  \
+\     (\\<forall>x. P x --> (M x & L x)) &   \
+\     ((\\<forall>x. P x --> Q x) | (\\<exists>x. P x & R x))  \
+\   --> (\\<exists>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));

 by (rtac go25 1);
 by (depth_prolog_tac horns25);
 
 
 writeln"Problem 26";
-Goal "((? 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))";
+Goal "((\\<exists>x. p x) = (\\<exists>x. q x)) &     \
+\     (\\<forall>x. \\<forall>y. p x & q y --> (r x = s y)) \
+\ --> ((\\<forall>x. p x --> r x) = (\\<forall>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));

 by (depth_prolog_tac horns26);  (*1.4 secs*)
 (*Proof is of length 107!!*)
 
 
 writeln"Problem 43  NOW PROVED AUTOMATICALLY!!";  (*16 Horn clauses*)
-Goal "(! x. ! y. q x y = (! z. p z x = (p z y::bool)))  \
-\     --> (! x. (! y. q x y = (q y x::bool)))";
+Goal "(\\<forall>x. \\<forall>y. q x y = (\\<forall>z. p z x = (p z y::bool)))  \
+\     --> (\\<forall>x. (\\<forall>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));

 #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)
+And here is the proof\\<forall> (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),

 by (meson_tac 1);
 result();
 
 writeln"Classical Logic: examples with quantifiers";
 
-Goal "(! x. P x & Q x) = ((! x. P x) & (! x. Q x))";
+Goal "(\\<forall>x. P x & Q x) = ((\\<forall>x. P x) & (\\<forall>x. Q x))";
 by (meson_tac 1);
 result(); 
 
-Goal "(? x. P --> Q x)  =  (P --> (? x. Q x))";
+Goal "(\\<exists>x. P --> Q x)  =  (P --> (\\<exists>x. Q x))";
 by (meson_tac 1);
 result(); 
 
-Goal "(? x. P x --> Q) = ((! x. P x) --> Q)";
+Goal "(\\<exists>x. P x --> Q) = ((\\<forall>x. P x) --> Q)";
 by (meson_tac 1);
 result(); 
 
-Goal "((! x. P x) | Q)  =  (! x. P x | Q)";
+Goal "((\\<forall>x. P x) | Q)  =  (\\<forall>x. P x | Q)";
 by (meson_tac 1);
 result(); 
 
-Goal "(! x. P x --> P(f x))  &  P d --> P(f(f(f d)))";
+Goal "(\\<forall>x. P x --> P(f x))  &  P d --> P(f(f(f d)))";
 by (meson_tac 1);
 result();
 
 (*Needs double instantiation of EXISTS*)
-Goal "? x. P x --> P a & P b";
-by (meson_tac 1);
-result();
-
-Goal "? z. P z --> (! x. P x)";
+Goal "\\<exists>x. P x --> P a & P b";
+by (meson_tac 1);
+result();
+
+Goal "\\<exists>z. P z --> (\\<forall>x. P x)";
 by (meson_tac 1);
 result();
 
 writeln"Hard examples with quantifiers";
 
 writeln"Problem 18";
-Goal "? y. ! x. P y --> P x";
+Goal "\\<exists>y. \\<forall>x. P y --> P x";
 by (meson_tac 1);
 result(); 
 
 writeln"Problem 19";
-Goal "? x. ! y z. (P y --> Q z) --> (P x --> Q x)";
+Goal "\\<exists>x. \\<forall>y z. (P y --> Q z) --> (P x --> Q x)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 20";
-Goal "(! x y. ? z. ! w. (P x & Q y --> R z & S w))     \
-\   --> (? x y. P x & Q y) --> (? z. R z)";
+Goal "(\\<forall>x y. \\<exists>z. \\<forall>w. (P x & Q y --> R z & S w))     \
+\   --> (\\<exists>x y. P x & Q y) --> (\\<exists>z. R z)";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 21";
-Goal "(? x. P --> Q x) & (? x. Q x --> P) --> (? x. P=Q x)";
+Goal "(\\<exists>x. P --> Q x) & (\\<exists>x. Q x --> P) --> (\\<exists>x. P=Q x)";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 22";
-Goal "(! x. P = Q x)  -->  (P = (! x. Q x))";
+Goal "(\\<forall>x. P = Q x)  -->  (P = (\\<forall>x. Q x))";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 23";
-Goal "(! x. P | Q x)  =  (P | (! x. Q x))";
+Goal "(\\<forall>x. P | Q x)  =  (P | (\\<forall>x. Q x))";
 by (meson_tac 1);  
 result();
 
 writeln"Problem 24";  (*The first goal clause is useless*)
-Goal "~(? 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)";
+Goal "~(\\<exists>x. S x & Q x) & (\\<forall>x. P x --> Q x | R x) &  \
+\     (~(\\<exists>x. P x) --> (\\<exists>x. Q x)) & (\\<forall>x. Q x | R x --> S x)  \
+\   --> (\\<exists>x. P x & R x)";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 25";
-Goal "(? 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)";
+Goal "(\\<exists>x. P x) &  \
+\     (\\<forall>x. L x --> ~ (M x & R x)) &  \
+\     (\\<forall>x. P x --> (M x & L x)) &   \
+\     ((\\<forall>x. P x --> Q x) | (\\<exists>x. P x & R x))  \
+\   --> (\\<exists>x. Q x & P x)";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 26";  (*24 Horn clauses*)
-Goal "((? 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))";
+Goal "((\\<exists>x. p x) = (\\<exists>x. q x)) &     \
+\     (\\<forall>x. \\<forall>y. p x & q y --> (r x = s y)) \
+\ --> ((\\<forall>x. p x --> r x) = (\\<forall>x. q x --> s x))";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 27";    (*13 Horn clauses*)
-Goal "(? 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)";
+Goal "(\\<exists>x. P x & ~Q x) &   \
+\     (\\<forall>x. P x --> R x) &   \
+\     (\\<forall>x. M x & L x --> P x) &   \
+\     ((\\<exists>x. R x & ~ Q x) --> (\\<forall>x. L x --> ~ R x))  \
+\     --> (\\<forall>x. M x --> ~L x)";
 by (meson_tac 1); 
 result();
 
 writeln"Problem 28.  AMENDED";  (*14 Horn clauses*)
-Goal "(! 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)";
+Goal "(\\<forall>x. P x --> (\\<forall>x. Q x)) &   \
+\     ((\\<forall>x. Q x | R x) --> (\\<exists>x. Q x & S x)) &  \
+\     ((\\<exists>x. S x) --> (\\<forall>x. L x --> M x))  \
+\   --> (\\<forall>x. P x & L x --> M x)";
 by (meson_tac 1);  
 result();
 
 writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
         (*62 Horn clauses*)
-Goal "(? 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))";
+Goal "(\\<exists>x. F x) & (\\<exists>y. G y)  \
+\   --> ( ((\\<forall>x. F x --> H x) & (\\<forall>y. G y --> J y))  =   \
+\         (\\<forall>x y. F x & G y --> H x & J y))";
 by (meson_tac 1);          (*0.7 secs*)
 result();
 
 writeln"Problem 30";
-Goal "(! x. P x | Q x --> ~ R x) & \
-\     (! x. (Q x --> ~ S x) --> P x & R x)  \
-\   --> (! x. S x)";
+Goal "(\\<forall>x. P x | Q x --> ~ R x) & \
+\     (\\<forall>x. (Q x --> ~ S x) --> P x & R x)  \
+\   --> (\\<forall>x. S x)";
 by (meson_tac 1);  
 result();
 
 writeln"Problem 31";  (*10 Horn clauses; first negative clauses is useless*)
-Goal "~(? x. P x & (Q x | R x)) & \
-\     (? x. L x & P x) & \
-\     (! x. ~ R x --> M x)  \
-\   --> (? x. L x & M x)";
+Goal "~(\\<exists>x. P x & (Q x | R x)) & \
+\     (\\<exists>x. L x & P x) & \
+\     (\\<forall>x. ~ R x --> M x)  \
+\   --> (\\<exists>x. L x & M x)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 32";
-Goal "(! 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)";
+Goal "(\\<forall>x. P x & (Q x | R x)-->S x) & \
+\     (\\<forall>x. S x & R x --> L x) & \
+\     (\\<forall>x. M x --> R x)  \
+\   --> (\\<forall>x. P x & M x --> L x)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 33";  (*55 Horn clauses*)
-Goal "(! x. P a & (P x --> P b)-->P c)  =    \
-\     (! x. (~P a | P x | P c) & (~P a | ~P b | P c))";
+Goal "(\\<forall>x. P a & (P x --> P b)-->P c)  =    \
+\     (\\<forall>x. (~P a | P x | P c) & (~P a | ~P b | P c))";
 by (meson_tac 1);          (*5.6 secs*)
 result();
 
 writeln"Problem 34  AMENDED (TWICE!!)"; (*924 Horn clauses*)
 (*Andrews's challenge*)
-Goal "((? x. ! y. p x = p y)  =               \
-\      ((? x. q x) = (! y. p y)))     =       \
-\     ((? x. ! y. q x = q y)  =               \
-\      ((? x. p x) = (! y. q y)))";
+Goal "((\\<exists>x. \\<forall>y. p x = p y)  =               \
+\      ((\\<exists>x. q x) = (\\<forall>y. p y)))     =       \
+\     ((\\<exists>x. \\<forall>y. q x = q y)  =               \
+\      ((\\<exists>x. p x) = (\\<forall>y. q y)))";
 by (meson_tac 1);          (*13 secs*)
 result();
 
 writeln"Problem 35";
-Goal "? x y. P x y -->  (! u v. P u v)";
+Goal "\\<exists>x y. P x y -->  (\\<forall>u v. P u v)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 36";  (*15 Horn clauses*)
-Goal "(! 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)";
+Goal "(\\<forall>x. \\<exists>y. J x y) & \
+\     (\\<forall>x. \\<exists>y. G x y) & \
+\     (\\<forall>x y. J x y | G x y -->       \
+\     (\\<forall>z. J y z | G y z --> H x z))   \
+\   --> (\\<forall>x. \\<exists>y. H x y)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 37";  (*10 Horn clauses*)
-Goal "(! 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)";
+Goal "(\\<forall>z. \\<exists>w. \\<forall>x. \\<exists>y. \
+\          (P x z --> P y w) & P y z & (P y w --> (\\<exists>u. Q u w))) & \
+\     (\\<forall>x z. ~P x z --> (\\<exists>y. Q y z)) & \
+\     ((\\<exists>x y. Q x y) --> (\\<forall>x. R x x))  \
+\   --> (\\<forall>x. \\<exists>y. R x y)";
 by (meson_tac 1);   (*causes unification tracing messages*)
 result();
 
 writeln"Problem 38";  (*Quite hard: 422 Horn clauses!!*)
-Goal "(! 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)))";
+Goal "(\\<forall>x. p a & (p x --> (\\<exists>y. p y & r x y)) -->            \
+\          (\\<exists>z. \\<exists>w. p z & r x w & r w z))  =                 \
+\     (\\<forall>x. (~p a | p x | (\\<exists>z. \\<exists>w. p z & r x w & r w z)) &  \
+\           (~p a | ~(\\<exists>y. p y & r x y) |                      \
+\            (\\<exists>z. \\<exists>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 "~ (? x. ! y. F y x = (~F y y))";
+Goal "~ (\\<exists>x. \\<forall>y. F y x = (~F y y))";
 by (meson_tac 1);
 result();
 
 writeln"Problem 40.  AMENDED";
-Goal "(? y. ! x. F x y = F x x)  \
-\     -->  ~ (! x. ? y. ! z. F z y = (~F z x))";
+Goal "(\\<exists>y. \\<forall>x. F x y = F x x)  \
+\     -->  ~ (\\<forall>x. \\<exists>y. \\<forall>z. F z y = (~F z x))";
 by (meson_tac 1);
 result();
 
 writeln"Problem 41";
-Goal "(! z. (? y. (! x. f x y = (f x z & ~ f x x))))    \
-\     --> ~ (? z. ! x. f x z)";
+Goal "(\\<forall>z. (\\<exists>y. (\\<forall>x. f x y = (f x z & ~ f x x))))    \
+\     --> ~ (\\<exists>z. \\<forall>x. f x z)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 42";
-Goal "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
+Goal "~ (\\<exists>y. \\<forall>x. p x y = (~ (\\<exists>z. p x z & p z x)))";
 by (meson_tac 1);
 result();
 
 writeln"Problem 43  NOW PROVED AUTOMATICALLY!!";
-Goal "(! x. ! y. q x y = (! z. p z x = (p z y::bool)))  \
-\     --> (! x. (! y. q x y = (q y x::bool)))";
+Goal "(\\<forall>x. \\<forall>y. q x y = (\\<forall>z. p z x = (p z y::bool)))  \
+\     --> (\\<forall>x. (\\<forall>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 "(! 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)";
+Goal "(\\<forall>x. f x -->                                    \
+\           (\\<exists>y. g y & h x y & (\\<exists>y. g y & ~ h x y)))  &   \
+\     (\\<exists>x. j x & (\\<forall>y. g y --> h x y))               \
+\     --> (\\<exists>x. j x & ~f x)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 45";  (*27 Horn clauses; 54-step proof*)
-Goal "(! 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))";
+Goal "(\\<forall>x. f x & (\\<forall>y. g y & h x y --> j x y)        \
+\           --> (\\<forall>y. g y & h x y --> k y)) &       \
+\     ~ (\\<exists>y. l y & k y) &                                    \
+\     (\\<exists>x. f x & (\\<forall>y. h x y --> l y)                        \
+\               & (\\<forall>y. g y & h x y --> j x y))             \
+\     --> (\\<exists>x. f x & ~ (\\<exists>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 "(! 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)";
+Goal "(\\<forall>x. f x & (\\<forall>y. f y & h y x --> g y) --> g x) &      \
+\     ((\\<exists>x. f x & ~g x) -->                                    \
+\     (\\<exists>x. f x & ~g x & (\\<forall>y. f y & ~g y --> j x y))) &    \
+\     (\\<forall>x y. f x & f y & h x y --> ~j y x)                    \
+\     --> (\\<forall>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
           87094 inferences so far.  Searching to depth 36*)
-Goal "(! 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))";
+Goal "(\\<forall>x. P1 x --> P0 x) & (\\<exists>x. P1 x) &     \
+\     (\\<forall>x. P2 x --> P0 x) & (\\<exists>x. P2 x) &     \
+\     (\\<forall>x. P3 x --> P0 x) & (\\<exists>x. P3 x) &     \
+\     (\\<forall>x. P4 x --> P0 x) & (\\<exists>x. P4 x) &     \
+\     (\\<forall>x. P5 x --> P0 x) & (\\<exists>x. P5 x) &     \
+\     (\\<forall>x. Q1 x --> Q0 x) & (\\<exists>x. Q1 x) &     \
+\     (\\<forall>x. P0 x --> ((\\<forall>y. Q0 y-->R x y) |    \
+\                      (\\<forall>y. P0 y & S y x &     \
+\                           (\\<exists>z. Q0 z&R y z) --> R x y))) &   \
+\     (\\<forall>x y. P3 y & (P5 x|P4 x) --> S x y) &        \
+\     (\\<forall>x y. P3 x & P2 y --> S x y) &        \
+\     (\\<forall>x y. P2 x & P1 y --> S x y) &        \
+\     (\\<forall>x y. P1 x & (P2 y|Q1 y) --> ~R x y) &       \
+\     (\\<forall>x y. P3 x & P4 y --> R x y) &        \
+\     (\\<forall>x y. P3 x & P5 y --> ~R x y) &       \
+\     (\\<forall>x. (P4 x|P5 x) --> (\\<exists>y. Q0 y & R x y))      \
+\     --> (\\<exists>x y. P0 x & P0 y & (\\<exists>z. Q1 z & R y z & R x y))";
 by (safe_best_meson_tac 1);     (*MUCH faster than iterative deepening*)
 result();
 
-(*The Los problem?  Circulated by John Harrison*)
-Goal "(! 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 y. P x y | Q x y)                           \
-\     --> (! x y. P x y) | (! x y. Q x y)";
+(*The Los problem\\<exists> Circulated by John Harrison*)
+Goal "(\\<forall>x y z. P x y & P y z --> P x z) &      \
+\     (\\<forall>x y z. Q x y & Q y z --> Q x z) &             \
+\     (\\<forall>x y. P x y --> P y x) &                       \
+\     (\\<forall>x y. P x y | Q x y)                           \
+\     --> (\\<forall>x y. P x y) | (\\<forall>x y. Q x y)";
 by (safe_best_meson_tac 1);     (*2.3 secs; iter. deepening is VERY slow*)
 result();
 
 (*A similar example, suggested by Johannes Schumann and credited to Pelletier*)
 Goal "(!x y z. P x y --> P y z --> P x z) --> \

 by (safe_best_meson_tac 1);          (*2.7 secs*)
 result();
 
 writeln"Problem 50";  
 (*What has this to do with equality?*)
-Goal "(! x. P a x | (! y. P x y)) --> (? x. ! y. P x y)";
+Goal "(\\<forall>x. P a x | (\\<forall>y. P x y)) --> (\\<exists>x. \\<forall>y. P x y)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 55";
 

 \     (!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)";
+\     (\\<exists>x. killed x agatha)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 57";
 Goal "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)";
+\     (\\<forall>x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 58";
 (* Challenge found on info-hol *)
-Goal "! P Q R x. ? v w. ! y z. P x & Q y --> (P v | R w) & (R z --> Q v)";
+Goal "\\<forall>P Q R x. \\<exists>v w. \\<forall>y z. P x & Q y --> (P v | R w) & (R z --> Q v)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 59";
-Goal "(! x. P x = (~P(f x))) --> (? x. P x & ~P(f x))";
+Goal "(\\<forall>x. P x = (~P(f x))) --> (\\<exists>x. P x & ~P(f x))";
 by (meson_tac 1);
 result();
 
 writeln"Problem 60";
-Goal "! x. P x (f x) = (? y. (! z. P z y --> P z (f x)) & P x y)";
+Goal "\\<forall>x. P x (f x) = (\\<exists>y. (\\<forall>z. P z y --> P z (f x)) & P x y)";
 by (meson_tac 1);
 result();
 
 writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
 Goal "(ALL x. p a & (p x --> p(f x)) --> p(f(f x)))  =   \

 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";
+val axa = "\\<forall>x y.   T(i x(i y x))";
+val axb = "\\<forall>x y z. T(i(i x(i y z))(i(i x y)(i x z)))";
+val axc = "\\<forall>x y.   T(i(i(n x)(n y))(i y x))";
+val axd = "\\<forall>x y.   T(i x y) & T x --> T y";
 
 fun axjoin ([],   q) = q
   | axjoin (p::ps, q) = "(" ^ p ^ ") --> (" ^ axjoin(ps,q) ^ ")";
 
-Goal (axjoin([axa,axb,axd], "! x. T(i x x)"));
+Goal (axjoin([axa,axb,axd], "\\<forall>x. T(i x x)"));
 by (meson_tac 1);  
 result();
 
 writeln"Problem 66";  
-Goal (axjoin([axa,axb,axc,axd], "! x. T(i x(n(n x)))"));
-(*TOO SLOW, several minutes!  
+Goal (axjoin([axa,axb,axc,axd], "\\<forall>x. T(i x(n(n x)))"));
+(*TOO SLOW, several minutes\\<forall> 
      208346 inferences so far.  Searching to depth 23
 by (meson_tac 1);
 result();
 *)
 
 writeln"Problem 67";  
-Goal (axjoin([axa,axb,axc,axd], "! x. T(i(n(n x)) x)"));
+Goal (axjoin([axa,axb,axc,axd], "\\<forall>x. T(i(n(n x)) x)"));
 (*TOO SLOW: more than 3 minutes!
   51061 inferences so far.  Searching to depth 21
 by (meson_tac 1);
 result();
 *)