src/Sequents/LK/hardquant.ML
author wenzelm
Sun Sep 18 15:20:08 2005 +0200 (2005-09-18)
changeset 17481 75166ebb619b
parent 7119 02d94b69ae04
permissions -rw-r--r--
converted to Isar theory format;
     1 (*  Title:      LK/ex/hard-quant
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 
     6 Hard examples with quantifiers.  Can be read to test the LK system.
     7 From  F. J. Pelletier,
     8   Seventy-Five Problems for Testing Automatic Theorem Provers,
     9   J. Automated Reasoning 2 (1986), 191-216.
    10   Errata, JAR 4 (1988), 236-236.
    11 
    12 Uses pc_tac rather than fast_tac when the former is significantly faster.
    13 *)
    14 
    15 context (theory "LK");
    16 
    17 Goal "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x))  &  (ALL x. Q(x))";
    18 by (Fast_tac 1);
    19 result();
    20 
    21 Goal "|- (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))";
    22 by (Fast_tac 1);
    23 result();
    24 
    25 Goal "|- (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q";
    26 by (Fast_tac 1);
    27 result();
    28 
    29 Goal "|- (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)";
    30 by (Fast_tac 1);
    31 result();
    32 
    33 writeln"Problems requiring quantifier duplication";
    34 
    35 (*Not provable by Fast_tac: needs multiple instantiation of ALL*)
    36 Goal "|- (ALL x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))";
    37 by (best_tac LK_dup_pack 1);
    38 result();
    39 
    40 (*Needs double instantiation of the quantifier*)
    41 Goal "|- EX x. P(x) --> P(a) & P(b)";
    42 by (fast_tac LK_dup_pack 1);
    43 result();
    44 
    45 Goal "|- EX z. P(z) --> (ALL x. P(x))";
    46 by (best_tac LK_dup_pack 1);
    47 result();
    48 
    49 writeln"Hard examples with quantifiers";
    50 
    51 writeln"Problem 18";
    52 Goal "|- EX y. ALL x. P(y)-->P(x)";
    53 by (best_tac LK_dup_pack 1);
    54 result();
    55 
    56 writeln"Problem 19";
    57 Goal "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
    58 by (best_tac LK_dup_pack 1);
    59 result();
    60 
    61 writeln"Problem 20";
    62 Goal "|- (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))     \
    63 \   --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
    64 by (Fast_tac 1);
    65 result();
    66 
    67 writeln"Problem 21";
    68 Goal "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
    69 by (best_tac LK_dup_pack 1);
    70 result();
    71 
    72 writeln"Problem 22";
    73 Goal "|- (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
    74 by (Fast_tac 1);
    75 result();
    76 
    77 writeln"Problem 23";
    78 Goal "|- (ALL x. P | Q(x))  <->  (P | (ALL x. Q(x)))";
    79 by (best_tac LK_pack 1);
    80 result();
    81 
    82 writeln"Problem 24";
    83 Goal "|- ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &  \
    84 \    ~(EX x. P(x)) --> (EX x. Q(x)) & (ALL x. Q(x)|R(x) --> S(x))  \
    85 \   --> (EX x. P(x)&R(x))";
    86 by (pc_tac LK_pack 1);
    87 result();
    88 
    89 writeln"Problem 25";
    90 Goal "|- (EX x. P(x)) &  \
    91 \       (ALL x. L(x) --> ~ (M(x) & R(x))) &  \
    92 \       (ALL x. P(x) --> (M(x) & L(x))) &   \
    93 \       ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))  \
    94 \   --> (EX x. Q(x)&P(x))";
    95 by (best_tac LK_pack 1);
    96 result();
    97 
    98 writeln"Problem 26";
    99 Goal "|- ((EX x. p(x)) <-> (EX x. q(x))) &       \
   100 \     (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y)))   \
   101 \ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))";
   102 by (pc_tac LK_pack 1);
   103 result();
   104 
   105 writeln"Problem 27";
   106 Goal "|- (EX x. P(x) & ~Q(x)) &   \
   107 \             (ALL x. P(x) --> R(x)) &   \
   108 \             (ALL x. M(x) & L(x) --> P(x)) &   \
   109 \             ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))  \
   110 \         --> (ALL x. M(x) --> ~L(x))";
   111 by (pc_tac LK_pack 1);
   112 result();
   113 
   114 writeln"Problem 28.  AMENDED";
   115 Goal "|- (ALL x. P(x) --> (ALL x. Q(x))) &   \
   116 \       ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &  \
   117 \       ((EX x. S(x)) --> (ALL x. L(x) --> M(x)))  \
   118 \   --> (ALL x. P(x) & L(x) --> M(x))";
   119 by (pc_tac LK_pack 1);
   120 result();
   121 
   122 writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
   123 Goal "|- (EX x. P(x)) & (EX y. Q(y))  \
   124 \   --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->     \
   125 \        (ALL x y. P(x) & Q(y) --> R(x) & S(y)))";
   126 by (pc_tac LK_pack 1);
   127 result();
   128 
   129 writeln"Problem 30";
   130 Goal "|- (ALL x. P(x) | Q(x) --> ~ R(x)) & \
   131 \       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
   132 \   --> (ALL x. S(x))";
   133 by (Fast_tac 1);
   134 result();
   135 
   136 writeln"Problem 31";
   137 Goal "|- ~(EX x. P(x) & (Q(x) | R(x))) & \
   138 \       (EX x. L(x) & P(x)) & \
   139 \       (ALL x. ~ R(x) --> M(x))  \
   140 \   --> (EX x. L(x) & M(x))";
   141 by (Fast_tac 1);
   142 result();
   143 
   144 writeln"Problem 32";
   145 Goal "|- (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
   146 \       (ALL x. S(x) & R(x) --> L(x)) & \
   147 \       (ALL x. M(x) --> R(x))  \
   148 \   --> (ALL x. P(x) & M(x) --> L(x))";
   149 by (best_tac LK_pack 1);
   150 result();
   151 
   152 writeln"Problem 33";
   153 Goal "|- (ALL x. P(a) & (P(x)-->P(b))-->P(c))  <->    \
   154 \    (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
   155 by (Fast_tac 1);
   156 result();
   157 
   158 writeln"Problem 34  AMENDED (TWICE!!)";
   159 (*Andrews's challenge*)
   160 Goal "|- ((EX x. ALL y. p(x) <-> p(y))  <->              \
   161 \              ((EX x. q(x)) <-> (ALL y. p(y))))     <->        \
   162 \             ((EX x. ALL y. q(x) <-> q(y))  <->                \
   163 \              ((EX x. p(x)) <-> (ALL y. q(y))))";
   164 by (best_tac LK_dup_pack 1); (*10 secs!*)
   165 result();
   166 
   167 
   168 writeln"Problem 35";
   169 Goal "|- EX x y. P(x,y) -->  (ALL u v. P(u,v))";
   170 by (best_tac LK_dup_pack 1);
   171 result();
   172 
   173 writeln"Problem 36";
   174 Goal "|- (ALL x. EX y. J(x,y)) & \
   175 \        (ALL x. EX y. G(x,y)) & \
   176 \        (ALL x y. J(x,y) | G(x,y) -->   \
   177 \        (ALL z. J(y,z) | G(y,z) --> H(x,z)))   \
   178 \        --> (ALL x. EX y. H(x,y))";
   179 by (Fast_tac 1);
   180 result();
   181 
   182 writeln"Problem 37";
   183 Goal "|- (ALL z. EX w. ALL x. EX y. \
   184 \          (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) & \
   185 \       (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \
   186 \       ((EX x y. Q(x,y)) --> (ALL x. R(x,x)))  \
   187 \   --> (ALL x. EX y. R(x,y))";
   188 by (pc_tac LK_pack 1);
   189 result();
   190 
   191 writeln"Problem 38";
   192 Goal "|- (ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) -->        \
   193 \                (EX z. EX w. p(z) & r(x,w) & r(w,z)))  <->         \
   194 \        (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) &    \
   195 \                (~p(a) | ~(EX y. p(y) & r(x,y)) |                          \
   196 \                (EX z. EX w. p(z) & r(x,w) & r(w,z))))";
   197 by (pc_tac LK_pack 1);
   198 result();
   199 
   200 writeln"Problem 39";
   201 Goal "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
   202 by (Fast_tac 1);
   203 result();
   204 
   205 writeln"Problem 40.  AMENDED";
   206 Goal "|- (EX y. ALL x. F(x,y) <-> F(x,x)) -->  \
   207 \        ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
   208 by (Fast_tac 1);
   209 result();
   210 
   211 writeln"Problem 41";
   212 Goal "|- (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x))      \
   213 \        --> ~ (EX z. ALL x. f(x,z))";
   214 by (Fast_tac 1);
   215 result();
   216 
   217 writeln"Problem 42";
   218 Goal "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";
   219 
   220 writeln"Problem 43  NOT PROVED AUTOMATICALLY";
   221 Goal "|- (ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \
   222 \         --> (ALL x. (ALL y. q(x,y) <-> q(y,x)))";
   223 
   224 writeln"Problem 44";
   225 Goal "|- (ALL x. f(x) -->                                        \
   226 \                (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &       \
   227 \        (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                   \
   228 \        --> (EX x. j(x) & ~f(x))";
   229 by (Fast_tac 1);
   230 result();
   231 
   232 writeln"Problem 45";
   233 Goal "|- (ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y))        \
   234 \                     --> (ALL y. g(y) & h(x,y) --> k(y))) &    \
   235 \     ~ (EX y. l(y) & k(y)) &                                   \
   236 \     (EX x. f(x) & (ALL y. h(x,y) --> l(y))                    \
   237 \                  & (ALL y. g(y) & h(x,y) --> j(x,y)))         \
   238 \     --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))";
   239 by (best_tac LK_pack 1);
   240 result();
   241 
   242 
   243 writeln"Problems (mainly) involving equality or functions";
   244 
   245 writeln"Problem 48";
   246 Goal "|- (a=b | c=d) & (a=c | b=d) --> a=d | b=c";
   247 by (fast_tac (pack() add_safes [subst]) 1);
   248 result();
   249 
   250 writeln"Problem 50";
   251 Goal "|- (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))";
   252 by (best_tac LK_dup_pack 1);
   253 result();
   254 
   255 writeln"Problem 51";
   256 Goal "|- (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
   257 \        (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
   258 by (fast_tac (pack() add_safes [subst]) 1);
   259 result();
   260 
   261 writeln"Problem 52";
   262 (*Almost the same as 51. *)
   263 Goal "|- (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->  \
   264 \        (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)";
   265 by (fast_tac (pack() add_safes [subst]) 1);
   266 result();
   267 
   268 writeln"Problem 56";
   269 Goal "|- (ALL x.(EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
   270 by (best_tac (pack() add_unsafes [symL, subst]) 1);
   271 (*requires tricker to orient the equality properly*)
   272 result();
   273 
   274 writeln"Problem 57";
   275 Goal "|- P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
   276 \        (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))";
   277 by (Fast_tac 1);
   278 result();
   279 
   280 writeln"Problem 58!";
   281 Goal "|- (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))";
   282 by (fast_tac (pack() add_safes [subst]) 1);
   283 result();
   284 
   285 writeln"Problem 59";
   286 (*Unification works poorly here -- the abstraction %sobj prevents efficient
   287   operation of the occurs check*)
   288 Unify.trace_bound := !Unify.search_bound - 1;
   289 Goal "|- (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
   290 by (best_tac LK_dup_pack 1);
   291 result();
   292 
   293 writeln"Problem 60";
   294 Goal "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
   295 by (Fast_tac 1);
   296 result();
   297 
   298 writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
   299 Goal "|- (ALL x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x))))  <->     \
   300 \     (ALL x. (~p(a) | p(x) | p(f(f(x)))) &                      \
   301 \             (~p(a) | ~p(f(x)) | p(f(f(x)))))";
   302 by (Fast_tac 1);
   303 result();
   304 
   305 (*18 June 92: loaded in 372 secs*)
   306 (*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*)
   307 (*29 June 92: loaded in 370 secs*)
   308 (*18 September 2005: loaded in 1.809 secs*)