src/FOLP/ex/Intuitionistic.thy
 author wenzelm Fri Apr 23 23:35:43 2010 +0200 (2010-04-23) changeset 36319 8feb2c4bef1a parent 35762 af3ff2ba4c54 child 58963 26bf09b95dda permissions -rw-r--r--
mark schematic statements explicitly;
```     1 (*  Title:      FOLP/ex/Intuitionistic.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1991  University of Cambridge
```
```     4
```
```     5 Intuitionistic First-Order Logic.
```
```     6
```
```     7 Single-step commands:
```
```     8 by (IntPr.step_tac 1)
```
```     9 by (biresolve_tac safe_brls 1);
```
```    10 by (biresolve_tac haz_brls 1);
```
```    11 by (assume_tac 1);
```
```    12 by (IntPr.safe_tac 1);
```
```    13 by (IntPr.mp_tac 1);
```
```    14 by (IntPr.fast_tac 1);
```
```    15 *)
```
```    16
```
```    17 (*Note: for PROPOSITIONAL formulae...
```
```    18   ~A is classically provable iff it is intuitionistically provable.
```
```    19   Therefore A is classically provable iff ~~A is intuitionistically provable.
```
```    20
```
```    21 Let Q be the conjuction of the propositions A|~A, one for each atom A in
```
```    22 P.  If P is provable classically, then clearly P&Q is provable
```
```    23 intuitionistically, so ~~(P&Q) is also provable intuitionistically.
```
```    24 The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
```
```    25 since ~~Q is intuitionistically provable.  Finally, if P is a negation then
```
```    26 ~~P is intuitionstically equivalent to P.  [Andy Pitts]
```
```    27 *)
```
```    28
```
```    29 theory Intuitionistic
```
```    30 imports IFOLP
```
```    31 begin
```
```    32
```
```    33 schematic_lemma "?p : ~~(P&Q) <-> ~~P & ~~Q"
```
```    34   by (tactic {* IntPr.fast_tac 1 *})
```
```    35
```
```    36 schematic_lemma "?p : ~~~P <-> ~P"
```
```    37   by (tactic {* IntPr.fast_tac 1 *})
```
```    38
```
```    39 schematic_lemma "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
```
```    40   by (tactic {* IntPr.fast_tac 1 *})
```
```    41
```
```    42 schematic_lemma "?p : (P<->Q) <-> (Q<->P)"
```
```    43   by (tactic {* IntPr.fast_tac 1 *})
```
```    44
```
```    45
```
```    46 subsection {* Lemmas for the propositional double-negation translation *}
```
```    47
```
```    48 schematic_lemma "?p : P --> ~~P"
```
```    49   by (tactic {* IntPr.fast_tac 1 *})
```
```    50
```
```    51 schematic_lemma "?p : ~~(~~P --> P)"
```
```    52   by (tactic {* IntPr.fast_tac 1 *})
```
```    53
```
```    54 schematic_lemma "?p : ~~P & ~~(P --> Q) --> ~~Q"
```
```    55   by (tactic {* IntPr.fast_tac 1 *})
```
```    56
```
```    57
```
```    58 subsection {* The following are classically but not constructively valid *}
```
```    59
```
```    60 (*The attempt to prove them terminates quickly!*)
```
```    61 schematic_lemma "?p : ((P-->Q) --> P)  -->  P"
```
```    62   apply (tactic {* IntPr.fast_tac 1 *})?
```
```    63   oops
```
```    64
```
```    65 schematic_lemma "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)"
```
```    66   apply (tactic {* IntPr.fast_tac 1 *})?
```
```    67   oops
```
```    68
```
```    69
```
```    70 subsection {* Intuitionistic FOL: propositional problems based on Pelletier *}
```
```    71
```
```    72 text "Problem ~~1"
```
```    73 schematic_lemma "?p : ~~((P-->Q)  <->  (~Q --> ~P))"
```
```    74   by (tactic {* IntPr.fast_tac 1 *})
```
```    75
```
```    76 text "Problem ~~2"
```
```    77 schematic_lemma "?p : ~~(~~P  <->  P)"
```
```    78   by (tactic {* IntPr.fast_tac 1 *})
```
```    79
```
```    80 text "Problem 3"
```
```    81 schematic_lemma "?p : ~(P-->Q) --> (Q-->P)"
```
```    82   by (tactic {* IntPr.fast_tac 1 *})
```
```    83
```
```    84 text "Problem ~~4"
```
```    85 schematic_lemma "?p : ~~((~P-->Q)  <->  (~Q --> P))"
```
```    86   by (tactic {* IntPr.fast_tac 1 *})
```
```    87
```
```    88 text "Problem ~~5"
```
```    89 schematic_lemma "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
```
```    90   by (tactic {* IntPr.fast_tac 1 *})
```
```    91
```
```    92 text "Problem ~~6"
```
```    93 schematic_lemma "?p : ~~(P | ~P)"
```
```    94   by (tactic {* IntPr.fast_tac 1 *})
```
```    95
```
```    96 text "Problem ~~7"
```
```    97 schematic_lemma "?p : ~~(P | ~~~P)"
```
```    98   by (tactic {* IntPr.fast_tac 1 *})
```
```    99
```
```   100 text "Problem ~~8.  Peirce's law"
```
```   101 schematic_lemma "?p : ~~(((P-->Q) --> P)  -->  P)"
```
```   102   by (tactic {* IntPr.fast_tac 1 *})
```
```   103
```
```   104 text "Problem 9"
```
```   105 schematic_lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
```
```   106   by (tactic {* IntPr.fast_tac 1 *})
```
```   107
```
```   108 text "Problem 10"
```
```   109 schematic_lemma "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
```
```   110   by (tactic {* IntPr.fast_tac 1 *})
```
```   111
```
```   112 text "11.  Proved in each direction (incorrectly, says Pelletier!!) "
```
```   113 schematic_lemma "?p : P<->P"
```
```   114   by (tactic {* IntPr.fast_tac 1 *})
```
```   115
```
```   116 text "Problem ~~12.  Dijkstra's law  "
```
```   117 schematic_lemma "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))"
```
```   118   by (tactic {* IntPr.fast_tac 1 *})
```
```   119
```
```   120 schematic_lemma "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))"
```
```   121   by (tactic {* IntPr.fast_tac 1 *})
```
```   122
```
```   123 text "Problem 13.  Distributive law"
```
```   124 schematic_lemma "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
```
```   125   by (tactic {* IntPr.fast_tac 1 *})
```
```   126
```
```   127 text "Problem ~~14"
```
```   128 schematic_lemma "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
```
```   129   by (tactic {* IntPr.fast_tac 1 *})
```
```   130
```
```   131 text "Problem ~~15"
```
```   132 schematic_lemma "?p : ~~((P --> Q) <-> (~P | Q))"
```
```   133   by (tactic {* IntPr.fast_tac 1 *})
```
```   134
```
```   135 text "Problem ~~16"
```
```   136 schematic_lemma "?p : ~~((P-->Q) | (Q-->P))"
```
```   137   by (tactic {* IntPr.fast_tac 1 *})
```
```   138
```
```   139 text "Problem ~~17"
```
```   140 schematic_lemma "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
```
```   141   by (tactic {* IntPr.fast_tac 1 *})  -- slow
```
```   142
```
```   143
```
```   144 subsection {* Examples with quantifiers *}
```
```   145
```
```   146 text "The converse is classical in the following implications..."
```
```   147
```
```   148 schematic_lemma "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
```
```   149   by (tactic {* IntPr.fast_tac 1 *})
```
```   150
```
```   151 schematic_lemma "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
```
```   152   by (tactic {* IntPr.fast_tac 1 *})
```
```   153
```
```   154 schematic_lemma "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
```
```   155   by (tactic {* IntPr.fast_tac 1 *})
```
```   156
```
```   157 schematic_lemma "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
```
```   158   by (tactic {* IntPr.fast_tac 1 *})
```
```   159
```
```   160 schematic_lemma "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
```
```   161   by (tactic {* IntPr.fast_tac 1 *})
```
```   162
```
```   163
```
```   164 text "The following are not constructively valid!"
```
```   165 text "The attempt to prove them terminates quickly!"
```
```   166
```
```   167 schematic_lemma "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
```
```   168   apply (tactic {* IntPr.fast_tac 1 *})?
```
```   169   oops
```
```   170
```
```   171 schematic_lemma "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
```
```   172   apply (tactic {* IntPr.fast_tac 1 *})?
```
```   173   oops
```
```   174
```
```   175 schematic_lemma "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
```
```   176   apply (tactic {* IntPr.fast_tac 1 *})?
```
```   177   oops
```
```   178
```
```   179 schematic_lemma "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
```
```   180   apply (tactic {* IntPr.fast_tac 1 *})?
```
```   181   oops
```
```   182
```
```   183 (*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
```
```   184 schematic_lemma "?p : EX x. Q(x) --> (ALL x. Q(x))"
```
```   185   apply (tactic {* IntPr.fast_tac 1 *})?
```
```   186   oops
```
```   187
```
```   188
```
```   189 subsection "Hard examples with quantifiers"
```
```   190
```
```   191 text {*
```
```   192   The ones that have not been proved are not known to be valid!
```
```   193   Some will require quantifier duplication -- not currently available.
```
```   194 *}
```
```   195
```
```   196 text "Problem ~~18"
```
```   197 schematic_lemma "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
```
```   198 (*NOT PROVED*)
```
```   199
```
```   200 text "Problem ~~19"
```
```   201 schematic_lemma "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
```
```   202 (*NOT PROVED*)
```
```   203
```
```   204 text "Problem 20"
```
```   205 schematic_lemma "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))
```
```   206     --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
```
```   207   by (tactic {* IntPr.fast_tac 1 *})
```
```   208
```
```   209 text "Problem 21"
```
```   210 schematic_lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
```
```   211 (*NOT PROVED*)
```
```   212
```
```   213 text "Problem 22"
```
```   214 schematic_lemma "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
```
```   215   by (tactic {* IntPr.fast_tac 1 *})
```
```   216
```
```   217 text "Problem ~~23"
```
```   218 schematic_lemma "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))"
```
```   219   by (tactic {* IntPr.fast_tac 1 *})
```
```   220
```
```   221 text "Problem 24"
```
```   222 schematic_lemma "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &
```
```   223      (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))
```
```   224     --> ~~(EX x. P(x)&R(x))"
```
```   225 (*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
```
```   226   apply (tactic "IntPr.safe_tac")
```
```   227   apply (erule impE)
```
```   228    apply (tactic "IntPr.fast_tac 1")
```
```   229   apply (tactic "IntPr.fast_tac 1")
```
```   230   done
```
```   231
```
```   232 text "Problem 25"
```
```   233 schematic_lemma "?p : (EX x. P(x)) &
```
```   234         (ALL x. L(x) --> ~ (M(x) & R(x))) &
```
```   235         (ALL x. P(x) --> (M(x) & L(x))) &
```
```   236         ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))
```
```   237     --> (EX x. Q(x)&P(x))"
```
```   238   by (tactic "IntPr.best_tac 1")
```
```   239
```
```   240 text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
```
```   241 schematic_lemma "?p : (EX x. P(x)) & (EX y. Q(y))
```
```   242     --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->
```
```   243          (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
```
```   244   by (tactic "IntPr.fast_tac 1")
```
```   245
```
```   246 text "Problem ~~30"
```
```   247 schematic_lemma "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) &
```
```   248         (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
```
```   249     --> (ALL x. ~~S(x))"
```
```   250   by (tactic "IntPr.fast_tac 1")
```
```   251
```
```   252 text "Problem 31"
```
```   253 schematic_lemma "?p : ~(EX x. P(x) & (Q(x) | R(x))) &
```
```   254         (EX x. L(x) & P(x)) &
```
```   255         (ALL x. ~ R(x) --> M(x))
```
```   256     --> (EX x. L(x) & M(x))"
```
```   257   by (tactic "IntPr.fast_tac 1")
```
```   258
```
```   259 text "Problem 32"
```
```   260 schematic_lemma "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &
```
```   261         (ALL x. S(x) & R(x) --> L(x)) &
```
```   262         (ALL x. M(x) --> R(x))
```
```   263     --> (ALL x. P(x) & M(x) --> L(x))"
```
```   264   by (tactic "IntPr.best_tac 1") -- slow
```
```   265
```
```   266 text "Problem 39"
```
```   267 schematic_lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
```
```   268   by (tactic "IntPr.best_tac 1")
```
```   269
```
```   270 text "Problem 40.  AMENDED"
```
```   271 schematic_lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->
```
```   272               ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
```
```   273   by (tactic "IntPr.best_tac 1") -- slow
```
```   274
```
```   275 text "Problem 44"
```
```   276 schematic_lemma "?p : (ALL x. f(x) -->
```
```   277               (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &
```
```   278               (EX x. j(x) & (ALL y. g(y) --> h(x,y)))
```
```   279               --> (EX x. j(x) & ~f(x))"
```
```   280   by (tactic "IntPr.best_tac 1")
```
```   281
```
```   282 text "Problem 48"
```
```   283 schematic_lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
```
```   284   by (tactic "IntPr.best_tac 1")
```
```   285
```
```   286 text "Problem 51"
```
```   287 schematic_lemma
```
```   288     "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->
```
```   289      (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"
```
```   290   by (tactic "IntPr.best_tac 1") -- {*60 seconds*}
```
```   291
```
```   292 text "Problem 56"
```
```   293 schematic_lemma "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
```
```   294   by (tactic "IntPr.best_tac 1")
```
```   295
```
```   296 text "Problem 57"
```
```   297 schematic_lemma
```
```   298     "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &
```
```   299      (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
```
```   300   by (tactic "IntPr.best_tac 1")
```
```   301
```
```   302 text "Problem 60"
```
```   303 schematic_lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
```
```   304   by (tactic "IntPr.best_tac 1")
```
```   305
```
```   306 end
```