src/FOLP/ex/Classical.thy

 theory Classical
 imports FOLP
 begin
 
 schematic_goal "?p : (P --> Q | R) --> (P-->Q) | (P-->R)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*If and only if*)
 schematic_goal "?p : (P<->Q) <-> (Q<->P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 schematic_goal "?p : ~ (P <-> ~P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 
 (*Sample problems from 
   F. J. Pelletier, 
   Seventy-Five Problems for Testing Automatic Theorem Provers,

 *)
 
 text "Pelletier's examples"
 (*1*)
 schematic_goal "?p : (P-->Q)  <->  (~Q --> ~P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*2*)
 schematic_goal "?p : ~ ~ P  <->  P"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*3*)
 schematic_goal "?p : ~(P-->Q) --> (Q-->P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*4*)
 schematic_goal "?p : (~P-->Q)  <->  (~Q --> P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*5*)
 schematic_goal "?p : ((P|Q)-->(P|R)) --> (P|(Q-->R))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*6*)
 schematic_goal "?p : P | ~ P"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*7*)
 schematic_goal "?p : P | ~ ~ ~ P"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*8.  Peirce's law*)
 schematic_goal "?p : ((P-->Q) --> P)  -->  P"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*9*)
 schematic_goal "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*10*)
 schematic_goal "?p : (Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
 schematic_goal "?p : P<->P"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*12.  "Dijkstra's law"*)
 schematic_goal "?p : ((P <-> Q) <-> R)  <->  (P <-> (Q <-> R))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*13.  Distributive law*)
 schematic_goal "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*14*)
 schematic_goal "?p : (P <-> Q) <-> ((Q | ~P) & (~Q|P))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*15*)
 schematic_goal "?p : (P --> Q) <-> (~P | Q)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*16*)
 schematic_goal "?p : (P-->Q) | (Q-->P)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 (*17*)
 schematic_goal "?p : ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 
 text "Classical Logic: examples with quantifiers"
 
 schematic_goal "?p : (ALL x. P(x) & Q(x)) <-> (ALL x. P(x))  &  (ALL x. Q(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 schematic_goal "?p : (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 schematic_goal "?p : (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 schematic_goal "?p : (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 
 text "Problems requiring quantifier duplication"
 
 (*Needs multiple instantiation of ALL.*)
 schematic_goal "?p : (ALL x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 (*Needs double instantiation of the quantifier*)
 schematic_goal "?p : EX x. P(x) --> P(a) & P(b)"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 schematic_goal "?p : EX z. P(z) --> (ALL x. P(x))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 
 text "Hard examples with quantifiers"
 
 text "Problem 18"
 schematic_goal "?p : EX y. ALL x. P(y)-->P(x)"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 19"
 schematic_goal "?p : EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 20"
 schematic_goal "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))      
     --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 21"
 schematic_goal "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 22"
 schematic_goal "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 23"
 schematic_goal "?p : (ALL x. P | Q(x))  <->  (P | (ALL x. Q(x)))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 24"
 schematic_goal "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &   
      (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))   
     --> (EX x. P(x)&R(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 25"
 schematic_goal "?p : (EX x. P(x)) &  
        (ALL x. L(x) --> ~ (M(x) & R(x))) &  
        (ALL x. P(x) --> (M(x) & L(x))) &   

 
 text "Problem 26"
 schematic_goal "?u : ((EX x. p(x)) <-> (EX x. q(x))) &   
      (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y)))   
   --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 27"
 schematic_goal "?p : (EX x. P(x) & ~Q(x)) &    
               (ALL x. P(x) --> R(x)) &    
               (ALL x. M(x) & L(x) --> P(x)) &    
               ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))   
           --> (ALL x. M(x) --> ~L(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 28.  AMENDED"
 schematic_goal "?p : (ALL x. P(x) --> (ALL x. Q(x))) &    
         ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &   
         ((EX x. S(x)) --> (ALL x. L(x) --> M(x)))   
     --> (ALL x. P(x) & L(x) --> M(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
 schematic_goal "?p : (EX x. P(x)) & (EX y. Q(y))   
     --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->      
          (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 30"
 schematic_goal "?p : (ALL x. P(x) | Q(x) --> ~ R(x)) &  
         (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))   
     --> (ALL x. S(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 31"
 schematic_goal "?p : ~(EX x. P(x) & (Q(x) | R(x))) &  
         (EX x. L(x) & P(x)) &  
         (ALL x. ~ R(x) --> M(x))   
     --> (EX x. L(x) & M(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 32"
 schematic_goal "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &  
         (ALL x. S(x) & R(x) --> L(x)) &  
         (ALL x. M(x) --> R(x))   
     --> (ALL x. P(x) & M(x) --> L(x))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 33"
 schematic_goal "?p : (ALL x. P(a) & (P(x)-->P(b))-->P(c))  <->     
      (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 35"
 schematic_goal "?p : EX x y. P(x,y) -->  (ALL u v. P(u,v))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 36"
 schematic_goal
 "?p : (ALL x. EX y. J(x,y)) &  
       (ALL x. EX y. G(x,y)) &  
       (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z)))    
   --> (ALL x. EX y. H(x,y))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 37"
 schematic_goal "?p : (ALL z. EX w. ALL x. EX y.  
            (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) &  
         (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) &  
         ((EX x y. Q(x,y)) --> (ALL x. R(x,x)))   
     --> (ALL x. EX y. R(x,y))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 39"
 schematic_goal "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 40.  AMENDED"
 schematic_goal "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->   
               ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 41"
 schematic_goal "?p : (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x))   
           --> ~ (EX z. ALL x. f(x,z))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 44"
 schematic_goal "?p : (ALL x. f(x) -->                                     
               (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &        
               (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                    
               --> (EX x. j(x) & ~f(x))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problems (mainly) involving equality or functions"
 
 text "Problem 48"
 schematic_goal "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 50"
 (*What has this to do with equality?*)
 schematic_goal "?p : (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 56"
 schematic_goal
  "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 57"
 schematic_goal
 "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &  
       (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 text "Problem 58  NOT PROVED AUTOMATICALLY"
 schematic_goal "?p : (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"
   supply f_cong = subst_context [where t = f]
-  by (tactic \<open>fast_tac @{context} (FOLP_cs addSIs [@{thm f_cong}]) 1\<close>)
+  by (tactic \<open>fast_tac \<^context> (FOLP_cs addSIs [@{thm f_cong}]) 1\<close>)
 
 text "Problem 59"
 schematic_goal "?p : (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"
-  by (tactic "best_tac @{context} FOLP_dup_cs 1")
+  by (tactic "best_tac \<^context> FOLP_dup_cs 1")
 
 text "Problem 60"
 schematic_goal "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
-  by (tactic "fast_tac @{context} FOLP_cs 1")
+  by (tactic "fast_tac \<^context> FOLP_cs 1")
 
 end