src/FOLP/ex/Intuitionistic.thy
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     1 (*  Title:      FOLP/ex/Intuitionistic.thy
       
     2     ID:         $Id$
       
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
       
     4     Copyright   1991  University of Cambridge
       
     5 
       
     6 Intuitionistic First-Order Logic.
       
     7 
       
     8 Single-step commands:
       
     9 by (IntPr.step_tac 1)
       
    10 by (biresolve_tac safe_brls 1);
       
    11 by (biresolve_tac haz_brls 1);
       
    12 by (assume_tac 1);
       
    13 by (IntPr.safe_tac 1);
       
    14 by (IntPr.mp_tac 1);
       
    15 by (IntPr.fast_tac 1);
       
    16 *)
       
    17 
       
    18 (*Note: for PROPOSITIONAL formulae...
       
    19   ~A is classically provable iff it is intuitionistically provable.  
       
    20   Therefore A is classically provable iff ~~A is intuitionistically provable.
       
    21 
       
    22 Let Q be the conjuction of the propositions A|~A, one for each atom A in
       
    23 P.  If P is provable classically, then clearly P&Q is provable
       
    24 intuitionistically, so ~~(P&Q) is also provable intuitionistically.
       
    25 The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
       
    26 since ~~Q is intuitionistically provable.  Finally, if P is a negation then
       
    27 ~~P is intuitionstically equivalent to P.  [Andy Pitts]
       
    28 *)
       
    29 
       
    30 theory Intuitionistic
       
    31 imports IFOLP
       
    32 begin
       
    33 
       
    34 lemma "?p : ~~(P&Q) <-> ~~P & ~~Q"
       
    35   by (tactic {* IntPr.fast_tac 1 *})
       
    36 
       
    37 lemma "?p : ~~~P <-> ~P"
       
    38   by (tactic {* IntPr.fast_tac 1 *})
       
    39 
       
    40 lemma "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
       
    41   by (tactic {* IntPr.fast_tac 1 *})
       
    42 
       
    43 lemma "?p : (P<->Q) <-> (Q<->P)"
       
    44   by (tactic {* IntPr.fast_tac 1 *})
       
    45 
       
    46 
       
    47 subsection {* Lemmas for the propositional double-negation translation *}
       
    48 
       
    49 lemma "?p : P --> ~~P"
       
    50   by (tactic {* IntPr.fast_tac 1 *})
       
    51 
       
    52 lemma "?p : ~~(~~P --> P)"
       
    53   by (tactic {* IntPr.fast_tac 1 *})
       
    54 
       
    55 lemma "?p : ~~P & ~~(P --> Q) --> ~~Q"
       
    56   by (tactic {* IntPr.fast_tac 1 *})
       
    57 
       
    58 
       
    59 subsection {* The following are classically but not constructively valid *}
       
    60 
       
    61 (*The attempt to prove them terminates quickly!*)
       
    62 lemma "?p : ((P-->Q) --> P)  -->  P"
       
    63   apply (tactic {* IntPr.fast_tac 1 *})?
       
    64   oops
       
    65 
       
    66 lemma "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)"
       
    67   apply (tactic {* IntPr.fast_tac 1 *})?
       
    68   oops
       
    69 
       
    70 
       
    71 subsection {* Intuitionistic FOL: propositional problems based on Pelletier *}
       
    72 
       
    73 text "Problem ~~1"
       
    74 lemma "?p : ~~((P-->Q)  <->  (~Q --> ~P))"
       
    75   by (tactic {* IntPr.fast_tac 1 *})
       
    76 
       
    77 text "Problem ~~2"
       
    78 lemma "?p : ~~(~~P  <->  P)"
       
    79   by (tactic {* IntPr.fast_tac 1 *})
       
    80 
       
    81 text "Problem 3"
       
    82 lemma "?p : ~(P-->Q) --> (Q-->P)"
       
    83   by (tactic {* IntPr.fast_tac 1 *})
       
    84 
       
    85 text "Problem ~~4"
       
    86 lemma "?p : ~~((~P-->Q)  <->  (~Q --> P))"
       
    87   by (tactic {* IntPr.fast_tac 1 *})
       
    88 
       
    89 text "Problem ~~5"
       
    90 lemma "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
       
    91   by (tactic {* IntPr.fast_tac 1 *})
       
    92 
       
    93 text "Problem ~~6"
       
    94 lemma "?p : ~~(P | ~P)"
       
    95   by (tactic {* IntPr.fast_tac 1 *})
       
    96 
       
    97 text "Problem ~~7"
       
    98 lemma "?p : ~~(P | ~~~P)"
       
    99   by (tactic {* IntPr.fast_tac 1 *})
       
   100 
       
   101 text "Problem ~~8.  Peirce's law"
       
   102 lemma "?p : ~~(((P-->Q) --> P)  -->  P)"
       
   103   by (tactic {* IntPr.fast_tac 1 *})
       
   104 
       
   105 text "Problem 9"
       
   106 lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
       
   107   by (tactic {* IntPr.fast_tac 1 *})
       
   108 
       
   109 text "Problem 10"
       
   110 lemma "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
       
   111   by (tactic {* IntPr.fast_tac 1 *})
       
   112 
       
   113 text "11.  Proved in each direction (incorrectly, says Pelletier!!) "
       
   114 lemma "?p : P<->P"
       
   115   by (tactic {* IntPr.fast_tac 1 *})
       
   116 
       
   117 text "Problem ~~12.  Dijkstra's law  "
       
   118 lemma "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))"
       
   119   by (tactic {* IntPr.fast_tac 1 *})
       
   120 
       
   121 lemma "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))"
       
   122   by (tactic {* IntPr.fast_tac 1 *})
       
   123 
       
   124 text "Problem 13.  Distributive law"
       
   125 lemma "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
       
   126   by (tactic {* IntPr.fast_tac 1 *})
       
   127 
       
   128 text "Problem ~~14"
       
   129 lemma "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
       
   130   by (tactic {* IntPr.fast_tac 1 *})
       
   131 
       
   132 text "Problem ~~15"
       
   133 lemma "?p : ~~((P --> Q) <-> (~P | Q))"
       
   134   by (tactic {* IntPr.fast_tac 1 *})
       
   135 
       
   136 text "Problem ~~16"
       
   137 lemma "?p : ~~((P-->Q) | (Q-->P))"
       
   138   by (tactic {* IntPr.fast_tac 1 *})
       
   139 
       
   140 text "Problem ~~17"
       
   141 lemma "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
       
   142   by (tactic {* IntPr.fast_tac 1 *})  -- slow
       
   143 
       
   144 
       
   145 subsection {* Examples with quantifiers *}
       
   146 
       
   147 text "The converse is classical in the following implications..."
       
   148 
       
   149 lemma "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
       
   150   by (tactic {* IntPr.fast_tac 1 *})
       
   151 
       
   152 lemma "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
       
   153   by (tactic {* IntPr.fast_tac 1 *})
       
   154 
       
   155 lemma "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
       
   156   by (tactic {* IntPr.fast_tac 1 *})
       
   157 
       
   158 lemma "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
       
   159   by (tactic {* IntPr.fast_tac 1 *})
       
   160 
       
   161 lemma "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
       
   162   by (tactic {* IntPr.fast_tac 1 *})
       
   163 
       
   164 
       
   165 text "The following are not constructively valid!"
       
   166 text "The attempt to prove them terminates quickly!"
       
   167 
       
   168 lemma "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
       
   169   apply (tactic {* IntPr.fast_tac 1 *})?
       
   170   oops
       
   171 
       
   172 lemma "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
       
   173   apply (tactic {* IntPr.fast_tac 1 *})?
       
   174   oops
       
   175 
       
   176 lemma "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
       
   177   apply (tactic {* IntPr.fast_tac 1 *})?
       
   178   oops
       
   179 
       
   180 lemma "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
       
   181   apply (tactic {* IntPr.fast_tac 1 *})?
       
   182   oops
       
   183 
       
   184 (*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
       
   185 lemma "?p : EX x. Q(x) --> (ALL x. Q(x))"
       
   186   apply (tactic {* IntPr.fast_tac 1 *})?
       
   187   oops
       
   188 
       
   189 
       
   190 subsection "Hard examples with quantifiers"
       
   191 
       
   192 text {*
       
   193   The ones that have not been proved are not known to be valid!
       
   194   Some will require quantifier duplication -- not currently available.
       
   195 *}
       
   196 
       
   197 text "Problem ~~18"
       
   198 lemma "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
       
   199 (*NOT PROVED*)
       
   200 
       
   201 text "Problem ~~19"
       
   202 lemma "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
       
   203 (*NOT PROVED*)
       
   204 
       
   205 text "Problem 20"
       
   206 lemma "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))      
       
   207     --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
       
   208   by (tactic {* IntPr.fast_tac 1 *})
       
   209 
       
   210 text "Problem 21"
       
   211 lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
       
   212 (*NOT PROVED*)
       
   213 
       
   214 text "Problem 22"
       
   215 lemma "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
       
   216   by (tactic {* IntPr.fast_tac 1 *})
       
   217 
       
   218 text "Problem ~~23"
       
   219 lemma "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))"
       
   220   by (tactic {* IntPr.fast_tac 1 *})
       
   221 
       
   222 text "Problem 24"
       
   223 lemma "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &   
       
   224      (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))   
       
   225     --> ~~(EX x. P(x)&R(x))"
       
   226 (*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
       
   227   apply (tactic "IntPr.safe_tac")
       
   228   apply (erule impE)
       
   229    apply (tactic "IntPr.fast_tac 1")
       
   230   apply (tactic "IntPr.fast_tac 1")
       
   231   done
       
   232 
       
   233 text "Problem 25"
       
   234 lemma "?p : (EX x. P(x)) &   
       
   235         (ALL x. L(x) --> ~ (M(x) & R(x))) &   
       
   236         (ALL x. P(x) --> (M(x) & L(x))) &    
       
   237         ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))   
       
   238     --> (EX x. Q(x)&P(x))"
       
   239   by (tactic "IntPr.best_tac 1")
       
   240 
       
   241 text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
       
   242 lemma "?p : (EX x. P(x)) & (EX y. Q(y))   
       
   243     --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->      
       
   244          (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
       
   245   by (tactic "IntPr.fast_tac 1")
       
   246 
       
   247 text "Problem ~~30"
       
   248 lemma "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) &  
       
   249         (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))   
       
   250     --> (ALL x. ~~S(x))"
       
   251   by (tactic "IntPr.fast_tac 1")
       
   252 
       
   253 text "Problem 31"
       
   254 lemma "?p : ~(EX x. P(x) & (Q(x) | R(x))) &  
       
   255         (EX x. L(x) & P(x)) &  
       
   256         (ALL x. ~ R(x) --> M(x))   
       
   257     --> (EX x. L(x) & M(x))"
       
   258   by (tactic "IntPr.fast_tac 1")
       
   259 
       
   260 text "Problem 32"
       
   261 lemma "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &  
       
   262         (ALL x. S(x) & R(x) --> L(x)) &  
       
   263         (ALL x. M(x) --> R(x))   
       
   264     --> (ALL x. P(x) & M(x) --> L(x))"
       
   265   by (tactic "IntPr.best_tac 1") -- slow
       
   266 
       
   267 text "Problem 39"
       
   268 lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
       
   269   by (tactic "IntPr.best_tac 1")
       
   270 
       
   271 text "Problem 40.  AMENDED"
       
   272 lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->   
       
   273               ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
       
   274   by (tactic "IntPr.best_tac 1") -- slow
       
   275 
       
   276 text "Problem 44"
       
   277 lemma "?p : (ALL x. f(x) -->                                    
       
   278               (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &        
       
   279               (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                    
       
   280               --> (EX x. j(x) & ~f(x))"
       
   281   by (tactic "IntPr.best_tac 1")
       
   282 
       
   283 text "Problem 48"
       
   284 lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
       
   285   by (tactic "IntPr.best_tac 1")
       
   286 
       
   287 text "Problem 51"
       
   288 lemma
       
   289     "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->   
       
   290      (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"
       
   291   by (tactic "IntPr.best_tac 1") -- {*60 seconds*}
       
   292 
       
   293 text "Problem 56"
       
   294 lemma "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
       
   295   by (tactic "IntPr.best_tac 1")
       
   296 
       
   297 text "Problem 57"
       
   298 lemma
       
   299     "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &  
       
   300      (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
       
   301   by (tactic "IntPr.best_tac 1")
       
   302 
       
   303 text "Problem 60"
       
   304 lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
       
   305   by (tactic "IntPr.best_tac 1")
       
   306 
       
   307 end