src/FOLP/ex/int.ML
 author clasohm Thu Sep 16 12:20:38 1993 +0200 (1993-09-16) changeset 0 a5a9c433f639 child 1459 d12da312eff4 permissions -rw-r--r--
Initial revision
```     1 (*  Title: 	FOL/ex/int
```
```     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 (Int.step_tac 1);
```
```    10 by (biresolve_tac safe_brls 1);
```
```    11 by (biresolve_tac haz_brls 1);
```
```    12 by (assume_tac 1);
```
```    13 by (Int.safe_tac 1);
```
```    14 by (Int.mp_tac 1);
```
```    15 by (Int.fast_tac 1);
```
```    16 *)
```
```    17
```
```    18 writeln"File FOL/ex/int.";
```
```    19
```
```    20 (*Note: for PROPOSITIONAL formulae...
```
```    21   ~A is classically provable iff it is intuitionistically provable.
```
```    22   Therefore A is classically provable iff ~~A is intuitionistically provable.
```
```    23
```
```    24 Let Q be the conjuction of the propositions A|~A, one for each atom A in
```
```    25 P.  If P is provable classically, then clearly P&Q is provable
```
```    26 intuitionistically, so ~~(P&Q) is also provable intuitionistically.
```
```    27 The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
```
```    28 since ~~Q is intuitionistically provable.  Finally, if P is a negation then
```
```    29 ~~P is intuitionstically equivalent to P.  [Andy Pitts]
```
```    30 *)
```
```    31
```
```    32 goal IFOLP.thy "?p : ~~(P&Q) <-> ~~P & ~~Q";
```
```    33 by (Int.fast_tac 1);
```
```    34 result();
```
```    35
```
```    36 goal IFOLP.thy "?p : ~~~P <-> ~P";
```
```    37 by (Int.fast_tac 1);
```
```    38 result();
```
```    39
```
```    40 goal IFOLP.thy "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))";
```
```    41 by (Int.fast_tac 1);
```
```    42 result();
```
```    43
```
```    44 goal IFOLP.thy "?p : (P<->Q) <-> (Q<->P)";
```
```    45 by (Int.fast_tac 1);
```
```    46 result();
```
```    47
```
```    48
```
```    49 writeln"Lemmas for the propositional double-negation translation";
```
```    50
```
```    51 goal IFOLP.thy "?p : P --> ~~P";
```
```    52 by (Int.fast_tac 1);
```
```    53 result();
```
```    54
```
```    55 goal IFOLP.thy "?p : ~~(~~P --> P)";
```
```    56 by (Int.fast_tac 1);
```
```    57 result();
```
```    58
```
```    59 goal IFOLP.thy "?p : ~~P & ~~(P --> Q) --> ~~Q";
```
```    60 by (Int.fast_tac 1);
```
```    61 result();
```
```    62
```
```    63
```
```    64 writeln"The following are classically but not constructively valid.";
```
```    65
```
```    66 (*The attempt to prove them terminates quickly!*)
```
```    67 goal IFOLP.thy "?p : ((P-->Q) --> P)  -->  P";
```
```    68 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```    69 (*Check that subgoals remain: proof failed.*)
```
```    70 getgoal 1;
```
```    71
```
```    72 goal IFOLP.thy "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)";
```
```    73 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```    74 getgoal 1;
```
```    75
```
```    76
```
```    77 writeln"Intuitionistic FOL: propositional problems based on Pelletier.";
```
```    78
```
```    79 writeln"Problem ~~1";
```
```    80 goal IFOLP.thy "?p : ~~((P-->Q)  <->  (~Q --> ~P))";
```
```    81 by (Int.fast_tac 1);
```
```    82 result();
```
```    83 (*5 secs*)
```
```    84
```
```    85
```
```    86 writeln"Problem ~~2";
```
```    87 goal IFOLP.thy "?p : ~~(~~P  <->  P)";
```
```    88 by (Int.fast_tac 1);
```
```    89 result();
```
```    90 (*1 secs*)
```
```    91
```
```    92
```
```    93 writeln"Problem 3";
```
```    94 goal IFOLP.thy "?p : ~(P-->Q) --> (Q-->P)";
```
```    95 by (Int.fast_tac 1);
```
```    96 result();
```
```    97
```
```    98 writeln"Problem ~~4";
```
```    99 goal IFOLP.thy "?p : ~~((~P-->Q)  <->  (~Q --> P))";
```
```   100 by (Int.fast_tac 1);
```
```   101 result();
```
```   102 (*9 secs*)
```
```   103
```
```   104 writeln"Problem ~~5";
```
```   105 goal IFOLP.thy "?p : ~~((P|Q-->P|R) --> P|(Q-->R))";
```
```   106 by (Int.fast_tac 1);
```
```   107 result();
```
```   108 (*10 secs*)
```
```   109
```
```   110
```
```   111 writeln"Problem ~~6";
```
```   112 goal IFOLP.thy "?p : ~~(P | ~P)";
```
```   113 by (Int.fast_tac 1);
```
```   114 result();
```
```   115
```
```   116 writeln"Problem ~~7";
```
```   117 goal IFOLP.thy "?p : ~~(P | ~~~P)";
```
```   118 by (Int.fast_tac 1);
```
```   119 result();
```
```   120
```
```   121 writeln"Problem ~~8.  Peirce's law";
```
```   122 goal IFOLP.thy "?p : ~~(((P-->Q) --> P)  -->  P)";
```
```   123 by (Int.fast_tac 1);
```
```   124 result();
```
```   125
```
```   126 writeln"Problem 9";
```
```   127 goal IFOLP.thy "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
```
```   128 by (Int.fast_tac 1);
```
```   129 result();
```
```   130 (*9 secs*)
```
```   131
```
```   132
```
```   133 writeln"Problem 10";
```
```   134 goal IFOLP.thy "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
```
```   135 by (Int.fast_tac 1);
```
```   136 result();
```
```   137
```
```   138 writeln"11.  Proved in each direction (incorrectly, says Pelletier!!) ";
```
```   139 goal IFOLP.thy "?p : P<->P";
```
```   140 by (Int.fast_tac 1);
```
```   141
```
```   142 writeln"Problem ~~12.  Dijkstra's law  ";
```
```   143 goal IFOLP.thy "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))";
```
```   144 by (Int.fast_tac 1);
```
```   145 result();
```
```   146
```
```   147 goal IFOLP.thy "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))";
```
```   148 by (Int.fast_tac 1);
```
```   149 result();
```
```   150
```
```   151 writeln"Problem 13.  Distributive law";
```
```   152 goal IFOLP.thy "?p : P | (Q & R)  <-> (P | Q) & (P | R)";
```
```   153 by (Int.fast_tac 1);
```
```   154 result();
```
```   155
```
```   156 writeln"Problem ~~14";
```
```   157 goal IFOLP.thy "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
```
```   158 by (Int.fast_tac 1);
```
```   159 result();
```
```   160
```
```   161 writeln"Problem ~~15";
```
```   162 goal IFOLP.thy "?p : ~~((P --> Q) <-> (~P | Q))";
```
```   163 by (Int.fast_tac 1);
```
```   164 result();
```
```   165
```
```   166 writeln"Problem ~~16";
```
```   167 goal IFOLP.thy "?p : ~~((P-->Q) | (Q-->P))";
```
```   168 by (Int.fast_tac 1);
```
```   169 result();
```
```   170
```
```   171 writeln"Problem ~~17";
```
```   172 goal IFOLP.thy
```
```   173   "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
```
```   174 by (Int.fast_tac 1);
```
```   175 (*over 5 minutes?? -- printing the proof term takes 40 secs!!*)
```
```   176 result();
```
```   177
```
```   178
```
```   179 writeln"** Examples with quantifiers **";
```
```   180
```
```   181 writeln"The converse is classical in the following implications...";
```
```   182
```
```   183 goal IFOLP.thy "?p : (EX x.P(x)-->Q)  -->  (ALL x.P(x)) --> Q";
```
```   184 by (Int.fast_tac 1);
```
```   185 result();
```
```   186
```
```   187 goal IFOLP.thy "?p : ((ALL x.P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
```
```   188 by (Int.fast_tac 1);
```
```   189 result();
```
```   190
```
```   191 goal IFOLP.thy "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))";
```
```   192 by (Int.fast_tac 1);
```
```   193 result();
```
```   194
```
```   195 goal IFOLP.thy "?p : (ALL x.P(x)) | Q  -->  (ALL x. P(x) | Q)";
```
```   196 by (Int.fast_tac 1);
```
```   197 result();
```
```   198
```
```   199 goal IFOLP.thy "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
```
```   200 by (Int.fast_tac 1);
```
```   201 result();
```
```   202
```
```   203
```
```   204
```
```   205
```
```   206 writeln"The following are not constructively valid!";
```
```   207 (*The attempt to prove them terminates quickly!*)
```
```   208
```
```   209 goal IFOLP.thy "?p : ((ALL x.P(x))-->Q) --> (EX x.P(x)-->Q)";
```
```   210 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```   211 getgoal 1;
```
```   212
```
```   213 goal IFOLP.thy "?p : (P --> (EX x.Q(x))) --> (EX x. P-->Q(x))";
```
```   214 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```   215 getgoal 1;
```
```   216
```
```   217 goal IFOLP.thy "?p : (ALL x. P(x) | Q) --> ((ALL x.P(x)) | Q)";
```
```   218 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```   219 getgoal 1;
```
```   220
```
```   221 goal IFOLP.thy "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
```
```   222 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```   223 getgoal 1;
```
```   224
```
```   225 (*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
```
```   226 goal IFOLP.thy "?p : EX x. Q(x) --> (ALL x. Q(x))";
```
```   227 by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
```
```   228 getgoal 1;
```
```   229
```
```   230
```
```   231 writeln"Hard examples with quantifiers";
```
```   232
```
```   233 (*The ones that have not been proved are not known to be valid!
```
```   234   Some will require quantifier duplication -- not currently available*)
```
```   235
```
```   236 writeln"Problem ~~18";
```
```   237 goal IFOLP.thy "?p : ~~(EX y. ALL x. P(y)-->P(x))";
```
```   238 (*NOT PROVED*)
```
```   239
```
```   240 writeln"Problem ~~19";
```
```   241 goal IFOLP.thy "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
```
```   242 (*NOT PROVED*)
```
```   243
```
```   244 writeln"Problem 20";
```
```   245 goal IFOLP.thy "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))     \
```
```   246 \   --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
```
```   247 by (Int.fast_tac 1);
```
```   248 result();
```
```   249
```
```   250 writeln"Problem 21";
```
```   251 goal IFOLP.thy
```
```   252     "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
```
```   253 (*NOT PROVED*)
```
```   254
```
```   255 writeln"Problem 22";
```
```   256 goal IFOLP.thy "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
```
```   257 by (Int.fast_tac 1);
```
```   258 result();
```
```   259
```
```   260 writeln"Problem ~~23";
```
```   261 goal IFOLP.thy "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))";
```
```   262 by (Int.best_tac 1);
```
```   263 result();
```
```   264
```
```   265 writeln"Problem 24";
```
```   266 goal IFOLP.thy "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &  \
```
```   267 \    ~(EX x.P(x)) --> (EX x.Q(x)) & (ALL x. Q(x)|R(x) --> S(x))  \
```
```   268 \   --> (EX x. P(x)&R(x))";
```
```   269 by (Int.fast_tac 1);
```
```   270 result();
```
```   271
```
```   272 writeln"Problem 25";
```
```   273 goal IFOLP.thy "?p : (EX x. P(x)) &  \
```
```   274 \       (ALL x. L(x) --> ~ (M(x) & R(x))) &  \
```
```   275 \       (ALL x. P(x) --> (M(x) & L(x))) &   \
```
```   276 \       ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))  \
```
```   277 \   --> (EX x. Q(x)&P(x))";
```
```   278 by (Int.best_tac 1);
```
```   279 result();
```
```   280
```
```   281 writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
```
```   282 goal IFOLP.thy "?p : (EX x. P(x)) & (EX y. Q(y))  \
```
```   283 \   --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->     \
```
```   284 \        (ALL x y. P(x) & Q(y) --> R(x) & S(y)))";
```
```   285 by (Int.fast_tac 1);
```
```   286 result();
```
```   287
```
```   288 writeln"Problem ~~30";
```
```   289 goal IFOLP.thy "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
```
```   290 \       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
```
```   291 \   --> (ALL x. ~~S(x))";
```
```   292 by (Int.fast_tac 1);
```
```   293 result();
```
```   294
```
```   295 writeln"Problem 31";
```
```   296 goal IFOLP.thy "?p : ~(EX x.P(x) & (Q(x) | R(x))) & \
```
```   297 \       (EX x. L(x) & P(x)) & \
```
```   298 \       (ALL x. ~ R(x) --> M(x))  \
```
```   299 \   --> (EX x. L(x) & M(x))";
```
```   300 by (Int.fast_tac 1);
```
```   301 result();
```
```   302
```
```   303 writeln"Problem 32";
```
```   304 goal IFOLP.thy "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
```
```   305 \       (ALL x. S(x) & R(x) --> L(x)) & \
```
```   306 \       (ALL x. M(x) --> R(x))  \
```
```   307 \   --> (ALL x. P(x) & M(x) --> L(x))";
```
```   308 by (Int.best_tac 1);  (*SLOW*)
```
```   309 result();
```
```   310
```
```   311 writeln"Problem 39";
```
```   312 goal IFOLP.thy "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
```
```   313 by (Int.fast_tac 1);
```
```   314 result();
```
```   315
```
```   316 writeln"Problem 40.  AMENDED";
```
```   317 goal IFOLP.thy "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->  \
```
```   318 \             ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
```
```   319 by (Int.fast_tac 1);
```
```   320 result();
```
```   321
```
```   322 writeln"Problem 44";
```
```   323 goal IFOLP.thy "?p : (ALL x. f(x) -->					\
```
```   324 \             (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &   	\
```
```   325 \             (EX x. j(x) & (ALL y. g(y) --> h(x,y)))			\
```
```   326 \             --> (EX x. j(x) & ~f(x))";
```
```   327 by (Int.fast_tac 1);
```
```   328 result();
```
```   329
```
```   330 writeln"Problem 48";
```
```   331 goal IFOLP.thy "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c";
```
```   332 by (Int.fast_tac 1);
```
```   333 result();
```
```   334
```
```   335 writeln"Problem 51";
```
```   336 goal IFOLP.thy
```
```   337     "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
```
```   338 \    (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
```
```   339 by (Int.best_tac 1);  (*60 seconds*)
```
```   340 result();
```
```   341
```
```   342 writeln"Problem 56";
```
```   343 goal IFOLP.thy
```
```   344     "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
```
```   345 by (Int.fast_tac 1);
```
```   346 result();
```
```   347
```
```   348 writeln"Problem 57";
```
```   349 goal IFOLP.thy
```
```   350     "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
```
```   351 \    (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))";
```
```   352 by (Int.fast_tac 1);
```
```   353 result();
```
```   354
```
```   355 writeln"Problem 60";
```
```   356 goal IFOLP.thy
```
```   357     "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
```
```   358 by (Int.fast_tac 1);
```
```   359 result();
```
```   360
```
```   361 writeln"Reached end of file.";
```