src/FOLP/ex/Intuitionistic.thy

 
 theory Intuitionistic
 imports IFOLP
 begin
 
-schematic_lemma "?p : ~~(P&Q) <-> ~~P & ~~Q"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ~~~P <-> ~P"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : (P<->Q) <-> (Q<->P)"
+schematic_goal "?p : ~~(P&Q) <-> ~~P & ~~Q"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ~~~P <-> ~P"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : (P<->Q) <-> (Q<->P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 
 subsection \<open>Lemmas for the propositional double-negation translation\<close>
 
-schematic_lemma "?p : P --> ~~P"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ~~(~~P --> P)"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ~~P & ~~(P --> Q) --> ~~Q"
+schematic_goal "?p : P --> ~~P"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ~~(~~P --> P)"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ~~P & ~~(P --> Q) --> ~~Q"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 
 subsection \<open>The following are classically but not constructively valid\<close>
 
 (*The attempt to prove them terminates quickly!*)
-schematic_lemma "?p : ((P-->Q) --> P)  -->  P"
-  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
-  oops
-
-schematic_lemma "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)"
+schematic_goal "?p : ((P-->Q) --> P)  -->  P"
+  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
+  oops
+
+schematic_goal "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)"
   apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
   oops
 
 
 subsection \<open>Intuitionistic FOL: propositional problems based on Pelletier\<close>
 
 text "Problem ~~1"
-schematic_lemma "?p : ~~((P-->Q)  <->  (~Q --> ~P))"
+schematic_goal "?p : ~~((P-->Q)  <->  (~Q --> ~P))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~2"
-schematic_lemma "?p : ~~(~~P  <->  P)"
+schematic_goal "?p : ~~(~~P  <->  P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 3"
-schematic_lemma "?p : ~(P-->Q) --> (Q-->P)"
+schematic_goal "?p : ~(P-->Q) --> (Q-->P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~4"
-schematic_lemma "?p : ~~((~P-->Q)  <->  (~Q --> P))"
+schematic_goal "?p : ~~((~P-->Q)  <->  (~Q --> P))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~5"
-schematic_lemma "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
+schematic_goal "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~6"
-schematic_lemma "?p : ~~(P | ~P)"
+schematic_goal "?p : ~~(P | ~P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~7"
-schematic_lemma "?p : ~~(P | ~~~P)"
+schematic_goal "?p : ~~(P | ~~~P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~8.  Peirce's law"
-schematic_lemma "?p : ~~(((P-->Q) --> P)  -->  P)"
+schematic_goal "?p : ~~(((P-->Q) --> P)  -->  P)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 9"
-schematic_lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
+schematic_goal "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 10"
-schematic_lemma "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
+schematic_goal "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "11.  Proved in each direction (incorrectly, says Pelletier!!) "
-schematic_lemma "?p : P<->P"
+schematic_goal "?p : P<->P"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~12.  Dijkstra's law  "
-schematic_lemma "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))"
+schematic_goal "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 13.  Distributive law"
-schematic_lemma "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
+schematic_goal "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~14"
-schematic_lemma "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
+schematic_goal "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~15"
-schematic_lemma "?p : ~~((P --> Q) <-> (~P | Q))"
+schematic_goal "?p : ~~((P --> Q) <-> (~P | Q))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~16"
-schematic_lemma "?p : ~~((P-->Q) | (Q-->P))"
+schematic_goal "?p : ~~((P-->Q) | (Q-->P))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~17"
-schematic_lemma "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
+schematic_goal "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)  -- slow
 
 
 subsection \<open>Examples with quantifiers\<close>
 
 text "The converse is classical in the following implications..."
 
-schematic_lemma "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
-  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
-
-schematic_lemma "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
+schematic_goal "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
+  by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
+
+schematic_goal "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 
 text "The following are not constructively valid!"
 text "The attempt to prove them terminates quickly!"
 
-schematic_lemma "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
-  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
-  oops
-
-schematic_lemma "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
-  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
-  oops
-
-schematic_lemma "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
-  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
-  oops
-
-schematic_lemma "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
+schematic_goal "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
+  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
+  oops
+
+schematic_goal "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
+  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
+  oops
+
+schematic_goal "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
+  apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
+  oops
+
+schematic_goal "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
   apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
   oops
 
 (*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
-schematic_lemma "?p : EX x. Q(x) --> (ALL x. Q(x))"
+schematic_goal "?p : EX x. Q(x) --> (ALL x. Q(x))"
   apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)?
   oops
 
 
 subsection "Hard examples with quantifiers"

   The ones that have not been proved are not known to be valid!
   Some will require quantifier duplication -- not currently available.
 \<close>
 
 text "Problem ~~18"
-schematic_lemma "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
+schematic_goal "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
 (*NOT PROVED*)
 
 text "Problem ~~19"
-schematic_lemma "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
+schematic_goal "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
 (*NOT PROVED*)
 
 text "Problem 20"
-schematic_lemma "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))      
+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 \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 21"
-schematic_lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
+schematic_goal "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
 (*NOT PROVED*)
 
 text "Problem 22"
-schematic_lemma "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
+schematic_goal "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem ~~23"
-schematic_lemma "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))"
+schematic_goal "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))"
   by (tactic \<open>IntPr.fast_tac @{context} 1\<close>)
 
 text "Problem 24"
-schematic_lemma "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &   
+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))"
 (*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
   apply (tactic "IntPr.safe_tac @{context}")
   apply (erule impE)
    apply (tactic "IntPr.fast_tac @{context} 1")
   apply (tactic "IntPr.fast_tac @{context} 1")
   done
 
 text "Problem 25"
-schematic_lemma "?p : (EX x. P(x)) &   
+schematic_goal "?p : (EX x. P(x)) &   
         (ALL x. L(x) --> ~ (M(x) & R(x))) &   
         (ALL x. P(x) --> (M(x) & L(x))) &    
         ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))   
     --> (EX x. Q(x)&P(x))"
   by (tactic "IntPr.best_tac @{context} 1")
 
 text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
-schematic_lemma "?p : (EX x. P(x)) & (EX y. Q(y))   
+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 "IntPr.fast_tac @{context} 1")
 
 text "Problem ~~30"
-schematic_lemma "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) &  
+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 "IntPr.fast_tac @{context} 1")
 
 text "Problem 31"
-schematic_lemma "?p : ~(EX x. P(x) & (Q(x) | R(x))) &  
+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 "IntPr.fast_tac @{context} 1")
 
 text "Problem 32"
-schematic_lemma "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &  
+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 "IntPr.best_tac @{context} 1") -- slow
 
 text "Problem 39"
-schematic_lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
+schematic_goal "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
   by (tactic "IntPr.best_tac @{context} 1")
 
 text "Problem 40.  AMENDED"
-schematic_lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->   
+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 "IntPr.best_tac @{context} 1") -- slow
 
 text "Problem 44"
-schematic_lemma "?p : (ALL x. f(x) -->                                    
+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 "IntPr.best_tac @{context} 1")
 
 text "Problem 48"
-schematic_lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
+schematic_goal "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
   by (tactic "IntPr.best_tac @{context} 1")
 
 text "Problem 51"
-schematic_lemma
+schematic_goal
     "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->   
      (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"
   by (tactic "IntPr.best_tac @{context} 1") -- \<open>60 seconds\<close>
 
 text "Problem 56"
-schematic_lemma "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
+schematic_goal "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
   by (tactic "IntPr.best_tac @{context} 1")
 
 text "Problem 57"
-schematic_lemma
+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 "IntPr.best_tac @{context} 1")
 
 text "Problem 60"
-schematic_lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
+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 "IntPr.best_tac @{context} 1")
 
 end