src/HOL/ex/Classical.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 18406 b1eab0eb7fec
child 21072 ede39342debf
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  Title:      HOL/ex/Classical
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 *)
     6 
     7 header{*Classical Predicate Calculus Problems*}
     8 
     9 theory Classical imports Main begin
    10 
    11 subsection{*Traditional Classical Reasoner*}
    12 
    13 text{*The machine "griffon" mentioned below is a 2.5GHz Power Mac G5.*}
    14 
    15 text{*Taken from @{text "FOL/Classical.thy"}. When porting examples from
    16 first-order logic, beware of the precedence of @{text "="} versus @{text
    17 "\<leftrightarrow>"}.*}
    18 
    19 lemma "(P --> Q | R) --> (P-->Q) | (P-->R)"
    20 by blast
    21 
    22 text{*If and only if*}
    23 
    24 lemma "(P=Q) = (Q = (P::bool))"
    25 by blast
    26 
    27 lemma "~ (P = (~P))"
    28 by blast
    29 
    30 
    31 text{*Sample problems from
    32   F. J. Pelletier,
    33   Seventy-Five Problems for Testing Automatic Theorem Provers,
    34   J. Automated Reasoning 2 (1986), 191-216.
    35   Errata, JAR 4 (1988), 236-236.
    36 
    37 The hardest problems -- judging by experience with several theorem provers,
    38 including matrix ones -- are 34 and 43.
    39 *}
    40 
    41 subsubsection{*Pelletier's examples*}
    42 
    43 text{*1*}
    44 lemma "(P-->Q)  =  (~Q --> ~P)"
    45 by blast
    46 
    47 text{*2*}
    48 lemma "(~ ~ P) =  P"
    49 by blast
    50 
    51 text{*3*}
    52 lemma "~(P-->Q) --> (Q-->P)"
    53 by blast
    54 
    55 text{*4*}
    56 lemma "(~P-->Q)  =  (~Q --> P)"
    57 by blast
    58 
    59 text{*5*}
    60 lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
    61 by blast
    62 
    63 text{*6*}
    64 lemma "P | ~ P"
    65 by blast
    66 
    67 text{*7*}
    68 lemma "P | ~ ~ ~ P"
    69 by blast
    70 
    71 text{*8.  Peirce's law*}
    72 lemma "((P-->Q) --> P)  -->  P"
    73 by blast
    74 
    75 text{*9*}
    76 lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
    77 by blast
    78 
    79 text{*10*}
    80 lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"
    81 by blast
    82 
    83 text{*11.  Proved in each direction (incorrectly, says Pelletier!!)  *}
    84 lemma "P=(P::bool)"
    85 by blast
    86 
    87 text{*12.  "Dijkstra's law"*}
    88 lemma "((P = Q) = R) = (P = (Q = R))"
    89 by blast
    90 
    91 text{*13.  Distributive law*}
    92 lemma "(P | (Q & R)) = ((P | Q) & (P | R))"
    93 by blast
    94 
    95 text{*14*}
    96 lemma "(P = Q) = ((Q | ~P) & (~Q|P))"
    97 by blast
    98 
    99 text{*15*}
   100 lemma "(P --> Q) = (~P | Q)"
   101 by blast
   102 
   103 text{*16*}
   104 lemma "(P-->Q) | (Q-->P)"
   105 by blast
   106 
   107 text{*17*}
   108 lemma "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))"
   109 by blast
   110 
   111 subsubsection{*Classical Logic: examples with quantifiers*}
   112 
   113 lemma "(\<forall>x. P(x) & Q(x)) = ((\<forall>x. P(x)) & (\<forall>x. Q(x)))"
   114 by blast
   115 
   116 lemma "(\<exists>x. P-->Q(x))  =  (P --> (\<exists>x. Q(x)))"
   117 by blast
   118 
   119 lemma "(\<exists>x. P(x)-->Q) = ((\<forall>x. P(x)) --> Q)"
   120 by blast
   121 
   122 lemma "((\<forall>x. P(x)) | Q)  =  (\<forall>x. P(x) | Q)"
   123 by blast
   124 
   125 text{*From Wishnu Prasetya*}
   126 lemma "(\<forall>s. q(s) --> r(s)) & ~r(s) & (\<forall>s. ~r(s) & ~q(s) --> p(t) | q(t))
   127     --> p(t) | r(t)"
   128 by blast
   129 
   130 
   131 subsubsection{*Problems requiring quantifier duplication*}
   132 
   133 text{*Theorem B of Peter Andrews, Theorem Proving via General Matings,
   134   JACM 28 (1981).*}
   135 lemma "(\<exists>x. \<forall>y. P(x) = P(y)) --> ((\<exists>x. P(x)) = (\<forall>y. P(y)))"
   136 by blast
   137 
   138 text{*Needs multiple instantiation of the quantifier.*}
   139 lemma "(\<forall>x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))"
   140 by blast
   141 
   142 text{*Needs double instantiation of the quantifier*}
   143 lemma "\<exists>x. P(x) --> P(a) & P(b)"
   144 by blast
   145 
   146 lemma "\<exists>z. P(z) --> (\<forall>x. P(x))"
   147 by blast
   148 
   149 lemma "\<exists>x. (\<exists>y. P(y)) --> P(x)"
   150 by blast
   151 
   152 subsubsection{*Hard examples with quantifiers*}
   153 
   154 text{*Problem 18*}
   155 lemma "\<exists>y. \<forall>x. P(y)-->P(x)"
   156 by blast
   157 
   158 text{*Problem 19*}
   159 lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
   160 by blast
   161 
   162 text{*Problem 20*}
   163 lemma "(\<forall>x y. \<exists>z. \<forall>w. (P(x)&Q(y)-->R(z)&S(w)))
   164     --> (\<exists>x y. P(x) & Q(y)) --> (\<exists>z. R(z))"
   165 by blast
   166 
   167 text{*Problem 21*}
   168 lemma "(\<exists>x. P-->Q(x)) & (\<exists>x. Q(x)-->P) --> (\<exists>x. P=Q(x))"
   169 by blast
   170 
   171 text{*Problem 22*}
   172 lemma "(\<forall>x. P = Q(x))  -->  (P = (\<forall>x. Q(x)))"
   173 by blast
   174 
   175 text{*Problem 23*}
   176 lemma "(\<forall>x. P | Q(x))  =  (P | (\<forall>x. Q(x)))"
   177 by blast
   178 
   179 text{*Problem 24*}
   180 lemma "~(\<exists>x. S(x)&Q(x)) & (\<forall>x. P(x) --> Q(x)|R(x)) &
   181      (~(\<exists>x. P(x)) --> (\<exists>x. Q(x))) & (\<forall>x. Q(x)|R(x) --> S(x))
   182     --> (\<exists>x. P(x)&R(x))"
   183 by blast
   184 
   185 text{*Problem 25*}
   186 lemma "(\<exists>x. P(x)) &
   187         (\<forall>x. L(x) --> ~ (M(x) & R(x))) &
   188         (\<forall>x. P(x) --> (M(x) & L(x))) &
   189         ((\<forall>x. P(x)-->Q(x)) | (\<exists>x. P(x)&R(x)))
   190     --> (\<exists>x. Q(x)&P(x))"
   191 by blast
   192 
   193 text{*Problem 26*}
   194 lemma "((\<exists>x. p(x)) = (\<exists>x. q(x))) &
   195       (\<forall>x. \<forall>y. p(x) & q(y) --> (r(x) = s(y)))
   196   --> ((\<forall>x. p(x)-->r(x)) = (\<forall>x. q(x)-->s(x)))"
   197 by blast
   198 
   199 text{*Problem 27*}
   200 lemma "(\<exists>x. P(x) & ~Q(x)) &
   201               (\<forall>x. P(x) --> R(x)) &
   202               (\<forall>x. M(x) & L(x) --> P(x)) &
   203               ((\<exists>x. R(x) & ~ Q(x)) --> (\<forall>x. L(x) --> ~ R(x)))
   204           --> (\<forall>x. M(x) --> ~L(x))"
   205 by blast
   206 
   207 text{*Problem 28.  AMENDED*}
   208 lemma "(\<forall>x. P(x) --> (\<forall>x. Q(x))) &
   209         ((\<forall>x. Q(x)|R(x)) --> (\<exists>x. Q(x)&S(x))) &
   210         ((\<exists>x. S(x)) --> (\<forall>x. L(x) --> M(x)))
   211     --> (\<forall>x. P(x) & L(x) --> M(x))"
   212 by blast
   213 
   214 text{*Problem 29.  Essentially the same as Principia Mathematica *11.71*}
   215 lemma "(\<exists>x. F(x)) & (\<exists>y. G(y))
   216     --> ( ((\<forall>x. F(x)-->H(x)) & (\<forall>y. G(y)-->J(y)))  =
   217           (\<forall>x y. F(x) & G(y) --> H(x) & J(y)))"
   218 by blast
   219 
   220 text{*Problem 30*}
   221 lemma "(\<forall>x. P(x) | Q(x) --> ~ R(x)) &
   222         (\<forall>x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
   223     --> (\<forall>x. S(x))"
   224 by blast
   225 
   226 text{*Problem 31*}
   227 lemma "~(\<exists>x. P(x) & (Q(x) | R(x))) &
   228         (\<exists>x. L(x) & P(x)) &
   229         (\<forall>x. ~ R(x) --> M(x))
   230     --> (\<exists>x. L(x) & M(x))"
   231 by blast
   232 
   233 text{*Problem 32*}
   234 lemma "(\<forall>x. P(x) & (Q(x)|R(x))-->S(x)) &
   235         (\<forall>x. S(x) & R(x) --> L(x)) &
   236         (\<forall>x. M(x) --> R(x))
   237     --> (\<forall>x. P(x) & M(x) --> L(x))"
   238 by blast
   239 
   240 text{*Problem 33*}
   241 lemma "(\<forall>x. P(a) & (P(x)-->P(b))-->P(c))  =
   242      (\<forall>x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
   243 by blast
   244 
   245 text{*Problem 34  AMENDED (TWICE!!)*}
   246 text{*Andrews's challenge*}
   247 lemma "((\<exists>x. \<forall>y. p(x) = p(y))  =
   248                ((\<exists>x. q(x)) = (\<forall>y. p(y))))   =
   249               ((\<exists>x. \<forall>y. q(x) = q(y))  =
   250                ((\<exists>x. p(x)) = (\<forall>y. q(y))))"
   251 by blast
   252 
   253 text{*Problem 35*}
   254 lemma "\<exists>x y. P x y -->  (\<forall>u v. P u v)"
   255 by blast
   256 
   257 text{*Problem 36*}
   258 lemma "(\<forall>x. \<exists>y. J x y) &
   259         (\<forall>x. \<exists>y. G x y) &
   260         (\<forall>x y. J x y | G x y -->
   261         (\<forall>z. J y z | G y z --> H x z))
   262     --> (\<forall>x. \<exists>y. H x y)"
   263 by blast
   264 
   265 text{*Problem 37*}
   266 lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
   267            (P x z -->P y w) & P y z & (P y w --> (\<exists>u. Q u w))) &
   268         (\<forall>x z. ~(P x z) --> (\<exists>y. Q y z)) &
   269         ((\<exists>x y. Q x y) --> (\<forall>x. R x x))
   270     --> (\<forall>x. \<exists>y. R x y)"
   271 by blast
   272 
   273 text{*Problem 38*}
   274 lemma "(\<forall>x. p(a) & (p(x) --> (\<exists>y. p(y) & r x y)) -->
   275            (\<exists>z. \<exists>w. p(z) & r x w & r w z))  =
   276      (\<forall>x. (~p(a) | p(x) | (\<exists>z. \<exists>w. p(z) & r x w & r w z)) &
   277            (~p(a) | ~(\<exists>y. p(y) & r x y) |
   278             (\<exists>z. \<exists>w. p(z) & r x w & r w z)))"
   279 by blast (*beats fast!*)
   280 
   281 text{*Problem 39*}
   282 lemma "~ (\<exists>x. \<forall>y. F y x = (~ F y y))"
   283 by blast
   284 
   285 text{*Problem 40.  AMENDED*}
   286 lemma "(\<exists>y. \<forall>x. F x y = F x x)
   287         -->  ~ (\<forall>x. \<exists>y. \<forall>z. F z y = (~ F z x))"
   288 by blast
   289 
   290 text{*Problem 41*}
   291 lemma "(\<forall>z. \<exists>y. \<forall>x. f x y = (f x z & ~ f x x))
   292                --> ~ (\<exists>z. \<forall>x. f x z)"
   293 by blast
   294 
   295 text{*Problem 42*}
   296 lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
   297 by blast
   298 
   299 text{*Problem 43!!*}
   300 lemma "(\<forall>x::'a. \<forall>y::'a. q x y = (\<forall>z. p z x = (p z y::bool)))
   301   --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
   302 by blast
   303 
   304 text{*Problem 44*}
   305 lemma "(\<forall>x. f(x) -->
   306               (\<exists>y. g(y) & h x y & (\<exists>y. g(y) & ~ h x y)))  &
   307               (\<exists>x. j(x) & (\<forall>y. g(y) --> h x y))
   308               --> (\<exists>x. j(x) & ~f(x))"
   309 by blast
   310 
   311 text{*Problem 45*}
   312 lemma "(\<forall>x. f(x) & (\<forall>y. g(y) & h x y --> j x y)
   313                       --> (\<forall>y. g(y) & h x y --> k(y))) &
   314      ~ (\<exists>y. l(y) & k(y)) &
   315      (\<exists>x. f(x) & (\<forall>y. h x y --> l(y))
   316                 & (\<forall>y. g(y) & h x y --> j x y))
   317       --> (\<exists>x. f(x) & ~ (\<exists>y. g(y) & h x y))"
   318 by blast
   319 
   320 
   321 subsubsection{*Problems (mainly) involving equality or functions*}
   322 
   323 text{*Problem 48*}
   324 lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
   325 by blast
   326 
   327 text{*Problem 49  NOT PROVED AUTOMATICALLY.
   328      Hard because it involves substitution for Vars
   329   the type constraint ensures that x,y,z have the same type as a,b,u. *}
   330 lemma "(\<exists>x y::'a. \<forall>z. z=x | z=y) & P(a) & P(b) & (~a=b)
   331                 --> (\<forall>u::'a. P(u))"
   332 apply safe
   333 apply (rule_tac x = a in allE, assumption)
   334 apply (rule_tac x = b in allE, assumption, fast)  --{*blast's treatment of equality can't do it*}
   335 done
   336 
   337 text{*Problem 50.  (What has this to do with equality?) *}
   338 lemma "(\<forall>x. P a x | (\<forall>y. P x y)) --> (\<exists>x. \<forall>y. P x y)"
   339 by blast
   340 
   341 text{*Problem 51*}
   342 lemma "(\<exists>z w. \<forall>x y. P x y = (x=z & y=w)) -->
   343      (\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P x y = (y=w)) = (x=z))"
   344 by blast
   345 
   346 text{*Problem 52. Almost the same as 51. *}
   347 lemma "(\<exists>z w. \<forall>x y. P x y = (x=z & y=w)) -->
   348      (\<exists>w. \<forall>y. \<exists>z. (\<forall>x. P x y = (x=z)) = (y=w))"
   349 by blast
   350 
   351 text{*Problem 55*}
   352 
   353 text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
   354   fast DISCOVERS who killed Agatha. *}
   355 lemma "lives(agatha) & lives(butler) & lives(charles) &
   356    (killed agatha agatha | killed butler agatha | killed charles agatha) &
   357    (\<forall>x y. killed x y --> hates x y & ~richer x y) &
   358    (\<forall>x. hates agatha x --> ~hates charles x) &
   359    (hates agatha agatha & hates agatha charles) &
   360    (\<forall>x. lives(x) & ~richer x agatha --> hates butler x) &
   361    (\<forall>x. hates agatha x --> hates butler x) &
   362    (\<forall>x. ~hates x agatha | ~hates x butler | ~hates x charles) -->
   363     killed ?who agatha"
   364 by fast
   365 
   366 text{*Problem 56*}
   367 lemma "(\<forall>x. (\<exists>y. P(y) & x=f(y)) --> P(x)) = (\<forall>x. P(x) --> P(f(x)))"
   368 by blast
   369 
   370 text{*Problem 57*}
   371 lemma "P (f a b) (f b c) & P (f b c) (f a c) &
   372      (\<forall>x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)"
   373 by blast
   374 
   375 text{*Problem 58  NOT PROVED AUTOMATICALLY*}
   376 lemma "(\<forall>x y. f(x)=g(y)) --> (\<forall>x y. f(f(x))=f(g(y)))"
   377 by (fast intro: arg_cong [of concl: f])
   378 
   379 text{*Problem 59*}
   380 lemma "(\<forall>x. P(x) = (~P(f(x)))) --> (\<exists>x. P(x) & ~P(f(x)))"
   381 by blast
   382 
   383 text{*Problem 60*}
   384 lemma "\<forall>x. P x (f x) = (\<exists>y. (\<forall>z. P z y --> P z (f x)) & P x y)"
   385 by blast
   386 
   387 text{*Problem 62 as corrected in JAR 18 (1997), page 135*}
   388 lemma "(\<forall>x. p a & (p x --> p(f x)) --> p(f(f x)))  =
   389       (\<forall>x. (~ p a | p x | p(f(f x))) &
   390               (~ p a | ~ p(f x) | p(f(f x))))"
   391 by blast
   392 
   393 text{*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
   394   fast indeed copes!*}
   395 lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
   396        (\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
   397        (\<forall>x. K(x) --> ~G(x))  -->  (\<exists>x. K(x) & J(x))"
   398 by fast
   399 
   400 text{*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
   401   It does seem obvious!*}
   402 lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
   403        (\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y)))  &
   404        (\<forall>x. K(x) --> ~G(x))   -->   (\<exists>x. K(x) --> ~G(x))"
   405 by fast
   406 
   407 text{*Attributed to Lewis Carroll by S. G. Pulman.  The first or last
   408 assumption can be deleted.*}
   409 lemma "(\<forall>x. honest(x) & industrious(x) --> healthy(x)) &
   410       ~ (\<exists>x. grocer(x) & healthy(x)) &
   411       (\<forall>x. industrious(x) & grocer(x) --> honest(x)) &
   412       (\<forall>x. cyclist(x) --> industrious(x)) &
   413       (\<forall>x. ~healthy(x) & cyclist(x) --> ~honest(x))
   414       --> (\<forall>x. grocer(x) --> ~cyclist(x))"
   415 by blast
   416 
   417 lemma "(\<forall>x y. R(x,y) | R(y,x)) &
   418        (\<forall>x y. S(x,y) & S(y,x) --> x=y) &
   419        (\<forall>x y. R(x,y) --> S(x,y))    -->   (\<forall>x y. S(x,y) --> R(x,y))"
   420 by blast
   421 
   422 
   423 subsection{*Model Elimination Prover*}
   424 
   425 
   426 text{*Trying out meson with arguments*}
   427 lemma "x < y & y < z --> ~ (z < (x::nat))"
   428 by (meson order_less_irrefl order_less_trans)
   429 
   430 text{*The "small example" from Bezem, Hendriks and de Nivelle,
   431 Automatic Proof Construction in Type Theory Using Resolution,
   432 JAR 29: 3-4 (2002), pages 253-275 *}
   433 lemma "(\<forall>x y z. R(x,y) & R(y,z) --> R(x,z)) &
   434        (\<forall>x. \<exists>y. R(x,y)) -->
   435        ~ (\<forall>x. P x = (\<forall>y. R(x,y) --> ~ P y))"
   436 by (tactic{*safe_best_meson_tac 1*})
   437     --{*In contrast, @{text meson} is SLOW: 7.6s on griffon*}
   438 
   439 
   440 subsubsection{*Pelletier's examples*}
   441 text{*1*}
   442 lemma "(P --> Q)  =  (~Q --> ~P)"
   443 by blast
   444 
   445 text{*2*}
   446 lemma "(~ ~ P) =  P"
   447 by blast
   448 
   449 text{*3*}
   450 lemma "~(P-->Q) --> (Q-->P)"
   451 by blast
   452 
   453 text{*4*}
   454 lemma "(~P-->Q)  =  (~Q --> P)"
   455 by blast
   456 
   457 text{*5*}
   458 lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
   459 by blast
   460 
   461 text{*6*}
   462 lemma "P | ~ P"
   463 by blast
   464 
   465 text{*7*}
   466 lemma "P | ~ ~ ~ P"
   467 by blast
   468 
   469 text{*8.  Peirce's law*}
   470 lemma "((P-->Q) --> P)  -->  P"
   471 by blast
   472 
   473 text{*9*}
   474 lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
   475 by blast
   476 
   477 text{*10*}
   478 lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"
   479 by blast
   480 
   481 text{*11.  Proved in each direction (incorrectly, says Pelletier!!)  *}
   482 lemma "P=(P::bool)"
   483 by blast
   484 
   485 text{*12.  "Dijkstra's law"*}
   486 lemma "((P = Q) = R) = (P = (Q = R))"
   487 by blast
   488 
   489 text{*13.  Distributive law*}
   490 lemma "(P | (Q & R)) = ((P | Q) & (P | R))"
   491 by blast
   492 
   493 text{*14*}
   494 lemma "(P = Q) = ((Q | ~P) & (~Q|P))"
   495 by blast
   496 
   497 text{*15*}
   498 lemma "(P --> Q) = (~P | Q)"
   499 by blast
   500 
   501 text{*16*}
   502 lemma "(P-->Q) | (Q-->P)"
   503 by blast
   504 
   505 text{*17*}
   506 lemma "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))"
   507 by blast
   508 
   509 subsubsection{*Classical Logic: examples with quantifiers*}
   510 
   511 lemma "(\<forall>x. P x & Q x) = ((\<forall>x. P x) & (\<forall>x. Q x))"
   512 by blast
   513 
   514 lemma "(\<exists>x. P --> Q x)  =  (P --> (\<exists>x. Q x))"
   515 by blast
   516 
   517 lemma "(\<exists>x. P x --> Q) = ((\<forall>x. P x) --> Q)"
   518 by blast
   519 
   520 lemma "((\<forall>x. P x) | Q)  =  (\<forall>x. P x | Q)"
   521 by blast
   522 
   523 lemma "(\<forall>x. P x --> P(f x))  &  P d --> P(f(f(f d)))"
   524 by blast
   525 
   526 text{*Needs double instantiation of EXISTS*}
   527 lemma "\<exists>x. P x --> P a & P b"
   528 by blast
   529 
   530 lemma "\<exists>z. P z --> (\<forall>x. P x)"
   531 by blast
   532 
   533 text{*From a paper by Claire Quigley*}
   534 lemma "\<exists>y. ((P c & Q y) | (\<exists>z. ~ Q z)) | (\<exists>x. ~ P x & Q d)"
   535 by fast
   536 
   537 subsubsection{*Hard examples with quantifiers*}
   538 
   539 text{*Problem 18*}
   540 lemma "\<exists>y. \<forall>x. P y --> P x"
   541 by blast
   542 
   543 text{*Problem 19*}
   544 lemma "\<exists>x. \<forall>y z. (P y --> Q z) --> (P x --> Q x)"
   545 by blast
   546 
   547 text{*Problem 20*}
   548 lemma "(\<forall>x y. \<exists>z. \<forall>w. (P x & Q y --> R z & S w))
   549     --> (\<exists>x y. P x & Q y) --> (\<exists>z. R z)"
   550 by blast
   551 
   552 text{*Problem 21*}
   553 lemma "(\<exists>x. P --> Q x) & (\<exists>x. Q x --> P) --> (\<exists>x. P=Q x)"
   554 by blast
   555 
   556 text{*Problem 22*}
   557 lemma "(\<forall>x. P = Q x)  -->  (P = (\<forall>x. Q x))"
   558 by blast
   559 
   560 text{*Problem 23*}
   561 lemma "(\<forall>x. P | Q x)  =  (P | (\<forall>x. Q x))"
   562 by blast
   563 
   564 text{*Problem 24*}  (*The first goal clause is useless*)
   565 lemma "~(\<exists>x. S x & Q x) & (\<forall>x. P x --> Q x | R x) &
   566       (~(\<exists>x. P x) --> (\<exists>x. Q x)) & (\<forall>x. Q x | R x --> S x)
   567     --> (\<exists>x. P x & R x)"
   568 by blast
   569 
   570 text{*Problem 25*}
   571 lemma "(\<exists>x. P x) &
   572       (\<forall>x. L x --> ~ (M x & R x)) &
   573       (\<forall>x. P x --> (M x & L x)) &
   574       ((\<forall>x. P x --> Q x) | (\<exists>x. P x & R x))
   575     --> (\<exists>x. Q x & P x)"
   576 by blast
   577 
   578 text{*Problem 26; has 24 Horn clauses*}
   579 lemma "((\<exists>x. p x) = (\<exists>x. q x)) &
   580       (\<forall>x. \<forall>y. p x & q y --> (r x = s y))
   581   --> ((\<forall>x. p x --> r x) = (\<forall>x. q x --> s x))"
   582 by blast
   583 
   584 text{*Problem 27; has 13 Horn clauses*}
   585 lemma "(\<exists>x. P x & ~Q x) &
   586       (\<forall>x. P x --> R x) &
   587       (\<forall>x. M x & L x --> P x) &
   588       ((\<exists>x. R x & ~ Q x) --> (\<forall>x. L x --> ~ R x))
   589       --> (\<forall>x. M x --> ~L x)"
   590 by blast
   591 
   592 text{*Problem 28.  AMENDED; has 14 Horn clauses*}
   593 lemma "(\<forall>x. P x --> (\<forall>x. Q x)) &
   594       ((\<forall>x. Q x | R x) --> (\<exists>x. Q x & S x)) &
   595       ((\<exists>x. S x) --> (\<forall>x. L x --> M x))
   596     --> (\<forall>x. P x & L x --> M x)"
   597 by blast
   598 
   599 text{*Problem 29.  Essentially the same as Principia Mathematica *11.71.
   600       62 Horn clauses*}
   601 lemma "(\<exists>x. F x) & (\<exists>y. G y)
   602     --> ( ((\<forall>x. F x --> H x) & (\<forall>y. G y --> J y))  =
   603           (\<forall>x y. F x & G y --> H x & J y))"
   604 by blast
   605 
   606 
   607 text{*Problem 30*}
   608 lemma "(\<forall>x. P x | Q x --> ~ R x) & (\<forall>x. (Q x --> ~ S x) --> P x & R x)
   609        --> (\<forall>x. S x)"
   610 by blast
   611 
   612 text{*Problem 31; has 10 Horn clauses; first negative clauses is useless*}
   613 lemma "~(\<exists>x. P x & (Q x | R x)) &
   614       (\<exists>x. L x & P x) &
   615       (\<forall>x. ~ R x --> M x)
   616     --> (\<exists>x. L x & M x)"
   617 by blast
   618 
   619 text{*Problem 32*}
   620 lemma "(\<forall>x. P x & (Q x | R x)-->S x) &
   621       (\<forall>x. S x & R x --> L x) &
   622       (\<forall>x. M x --> R x)
   623     --> (\<forall>x. P x & M x --> L x)"
   624 by blast
   625 
   626 text{*Problem 33; has 55 Horn clauses*}
   627 lemma "(\<forall>x. P a & (P x --> P b)-->P c)  =
   628       (\<forall>x. (~P a | P x | P c) & (~P a | ~P b | P c))"
   629 by blast
   630 
   631 text{*Problem 34: Andrews's challenge has 924 Horn clauses*}
   632 lemma "((\<exists>x. \<forall>y. p x = p y)  = ((\<exists>x. q x) = (\<forall>y. p y)))     =
   633       ((\<exists>x. \<forall>y. q x = q y)  = ((\<exists>x. p x) = (\<forall>y. q y)))"
   634 by blast
   635 
   636 text{*Problem 35*}
   637 lemma "\<exists>x y. P x y -->  (\<forall>u v. P u v)"
   638 by blast
   639 
   640 text{*Problem 36; has 15 Horn clauses*}
   641 lemma "(\<forall>x. \<exists>y. J x y) & (\<forall>x. \<exists>y. G x y) &
   642        (\<forall>x y. J x y | G x y --> (\<forall>z. J y z | G y z --> H x z))
   643        --> (\<forall>x. \<exists>y. H x y)"
   644 by blast
   645 
   646 text{*Problem 37; has 10 Horn clauses*}
   647 lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
   648            (P x z --> P y w) & P y z & (P y w --> (\<exists>u. Q u w))) &
   649       (\<forall>x z. ~P x z --> (\<exists>y. Q y z)) &
   650       ((\<exists>x y. Q x y) --> (\<forall>x. R x x))
   651     --> (\<forall>x. \<exists>y. R x y)"
   652 by blast --{*causes unification tracing messages*}
   653 
   654 
   655 text{*Problem 38*}  text{*Quite hard: 422 Horn clauses!!*}
   656 lemma "(\<forall>x. p a & (p x --> (\<exists>y. p y & r x y)) -->
   657            (\<exists>z. \<exists>w. p z & r x w & r w z))  =
   658       (\<forall>x. (~p a | p x | (\<exists>z. \<exists>w. p z & r x w & r w z)) &
   659             (~p a | ~(\<exists>y. p y & r x y) |
   660              (\<exists>z. \<exists>w. p z & r x w & r w z)))"
   661 by blast
   662 
   663 text{*Problem 39*}
   664 lemma "~ (\<exists>x. \<forall>y. F y x = (~F y y))"
   665 by blast
   666 
   667 text{*Problem 40.  AMENDED*}
   668 lemma "(\<exists>y. \<forall>x. F x y = F x x)
   669       -->  ~ (\<forall>x. \<exists>y. \<forall>z. F z y = (~F z x))"
   670 by blast
   671 
   672 text{*Problem 41*}
   673 lemma "(\<forall>z. (\<exists>y. (\<forall>x. f x y = (f x z & ~ f x x))))
   674       --> ~ (\<exists>z. \<forall>x. f x z)"
   675 by blast
   676 
   677 text{*Problem 42*}
   678 lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
   679 by blast
   680 
   681 text{*Problem 43  NOW PROVED AUTOMATICALLY!!*}
   682 lemma "(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool)))
   683       --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
   684 by blast
   685 
   686 text{*Problem 44: 13 Horn clauses; 7-step proof*}
   687 lemma "(\<forall>x. f x --> (\<exists>y. g y & h x y & (\<exists>y. g y & ~ h x y)))  &
   688        (\<exists>x. j x & (\<forall>y. g y --> h x y))
   689        --> (\<exists>x. j x & ~f x)"
   690 by blast
   691 
   692 text{*Problem 45; has 27 Horn clauses; 54-step proof*}
   693 lemma "(\<forall>x. f x & (\<forall>y. g y & h x y --> j x y)
   694             --> (\<forall>y. g y & h x y --> k y)) &
   695       ~ (\<exists>y. l y & k y) &
   696       (\<exists>x. f x & (\<forall>y. h x y --> l y)
   697                 & (\<forall>y. g y & h x y --> j x y))
   698       --> (\<exists>x. f x & ~ (\<exists>y. g y & h x y))"
   699 by blast
   700 
   701 text{*Problem 46; has 26 Horn clauses; 21-step proof*}
   702 lemma "(\<forall>x. f x & (\<forall>y. f y & h y x --> g y) --> g x) &
   703        ((\<exists>x. f x & ~g x) -->
   704        (\<exists>x. f x & ~g x & (\<forall>y. f y & ~g y --> j x y))) &
   705        (\<forall>x y. f x & f y & h x y --> ~j y x)
   706        --> (\<forall>x. f x --> g x)"
   707 by blast
   708 
   709 text{*Problem 47.  Schubert's Steamroller.
   710       26 clauses; 63 Horn clauses.
   711       87094 inferences so far.  Searching to depth 36*}
   712 lemma "(\<forall>x. wolf x \<longrightarrow> animal x) & (\<exists>x. wolf x) &
   713        (\<forall>x. fox x \<longrightarrow> animal x) & (\<exists>x. fox x) &
   714        (\<forall>x. bird x \<longrightarrow> animal x) & (\<exists>x. bird x) &
   715        (\<forall>x. caterpillar x \<longrightarrow> animal x) & (\<exists>x. caterpillar x) &
   716        (\<forall>x. snail x \<longrightarrow> animal x) & (\<exists>x. snail x) &
   717        (\<forall>x. grain x \<longrightarrow> plant x) & (\<exists>x. grain x) &
   718        (\<forall>x. animal x \<longrightarrow>
   719              ((\<forall>y. plant y \<longrightarrow> eats x y)  \<or> 
   720 	      (\<forall>y. animal y & smaller_than y x &
   721                     (\<exists>z. plant z & eats y z) \<longrightarrow> eats x y))) &
   722        (\<forall>x y. bird y & (snail x \<or> caterpillar x) \<longrightarrow> smaller_than x y) &
   723        (\<forall>x y. bird x & fox y \<longrightarrow> smaller_than x y) &
   724        (\<forall>x y. fox x & wolf y \<longrightarrow> smaller_than x y) &
   725        (\<forall>x y. wolf x & (fox y \<or> grain y) \<longrightarrow> ~eats x y) &
   726        (\<forall>x y. bird x & caterpillar y \<longrightarrow> eats x y) &
   727        (\<forall>x y. bird x & snail y \<longrightarrow> ~eats x y) &
   728        (\<forall>x. (caterpillar x \<or> snail x) \<longrightarrow> (\<exists>y. plant y & eats x y))
   729        \<longrightarrow> (\<exists>x y. animal x & animal y & (\<exists>z. grain z & eats y z & eats x y))"
   730 by (tactic{*safe_best_meson_tac 1*})
   731     --{*Nearly twice as fast as @{text meson},
   732         which performs iterative deepening rather than best-first search*}
   733 
   734 text{*The Los problem. Circulated by John Harrison*}
   735 lemma "(\<forall>x y z. P x y & P y z --> P x z) &
   736        (\<forall>x y z. Q x y & Q y z --> Q x z) &
   737        (\<forall>x y. P x y --> P y x) &
   738        (\<forall>x y. P x y | Q x y)
   739        --> (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
   740 by meson
   741 
   742 text{*A similar example, suggested by Johannes Schumann and
   743  credited to Pelletier*}
   744 lemma "(\<forall>x y z. P x y --> P y z --> P x z) -->
   745        (\<forall>x y z. Q x y --> Q y z --> Q x z) -->
   746        (\<forall>x y. Q x y --> Q y x) -->  (\<forall>x y. P x y | Q x y) -->
   747        (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
   748 by meson
   749 
   750 text{*Problem 50.  What has this to do with equality?*}
   751 lemma "(\<forall>x. P a x | (\<forall>y. P x y)) --> (\<exists>x. \<forall>y. P x y)"
   752 by blast
   753 
   754 text{*Problem 54: NOT PROVED*}
   755 lemma "(\<forall>y::'a. \<exists>z. \<forall>x. F x z = (x=y)) -->
   756       ~ (\<exists>w. \<forall>x. F x w = (\<forall>u. F x u --> (\<exists>y. F y u & ~ (\<exists>z. F z u & F z y))))"
   757 oops 
   758 
   759 
   760 text{*Problem 55*}
   761 
   762 text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
   763   @{text meson} cannot report who killed Agatha. *}
   764 lemma "lives agatha & lives butler & lives charles &
   765        (killed agatha agatha | killed butler agatha | killed charles agatha) &
   766        (\<forall>x y. killed x y --> hates x y & ~richer x y) &
   767        (\<forall>x. hates agatha x --> ~hates charles x) &
   768        (hates agatha agatha & hates agatha charles) &
   769        (\<forall>x. lives x & ~richer x agatha --> hates butler x) &
   770        (\<forall>x. hates agatha x --> hates butler x) &
   771        (\<forall>x. ~hates x agatha | ~hates x butler | ~hates x charles) -->
   772        (\<exists>x. killed x agatha)"
   773 by meson
   774 
   775 text{*Problem 57*}
   776 lemma "P (f a b) (f b c) & P (f b c) (f a c) &
   777       (\<forall>x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)"
   778 by blast
   779 
   780 text{*Problem 58: Challenge found on info-hol *}
   781 lemma "\<forall>P Q R x. \<exists>v w. \<forall>y z. P x & Q y --> (P v | R w) & (R z --> Q v)"
   782 by blast
   783 
   784 text{*Problem 59*}
   785 lemma "(\<forall>x. P x = (~P(f x))) --> (\<exists>x. P x & ~P(f x))"
   786 by blast
   787 
   788 text{*Problem 60*}
   789 lemma "\<forall>x. P x (f x) = (\<exists>y. (\<forall>z. P z y --> P z (f x)) & P x y)"
   790 by blast
   791 
   792 text{*Problem 62 as corrected in JAR 18 (1997), page 135*}
   793 lemma "(\<forall>x. p a & (p x --> p(f x)) --> p(f(f x)))  =
   794        (\<forall>x. (~ p a | p x | p(f(f x))) &
   795             (~ p a | ~ p(f x) | p(f(f x))))"
   796 by blast
   797 
   798 text{** Charles Morgan's problems **}
   799 
   800 lemma
   801   assumes a: "\<forall>x y.  T(i x(i y x))"
   802       and b: "\<forall>x y z. T(i (i x (i y z)) (i (i x y) (i x z)))"
   803       and c: "\<forall>x y.   T(i (i (n x) (n y)) (i y x))"
   804       and c': "\<forall>x y.   T(i (i y x) (i (n x) (n y)))"
   805       and d: "\<forall>x y.   T(i x y) & T x --> T y"
   806  shows True
   807 proof -
   808   from a b d have "\<forall>x. T(i x x)" by blast
   809   from a b c d have "\<forall>x. T(i x (n(n x)))" --{*Problem 66*}
   810     by meson
   811       --{*SLOW: 18s on griffon. 208346 inferences, depth 23 *}
   812   from a b c d have "\<forall>x. T(i (n(n x)) x)" --{*Problem 67*}
   813     by meson
   814       --{*4.9s on griffon. 51061 inferences, depth 21 *}
   815   from a b c' d have "\<forall>x. T(i x (n(n x)))" 
   816       --{*Problem 68: not proved.  Listed as satisfiable in TPTP (LCL078-1)*}
   817 oops
   818 
   819 text{*Problem 71, as found in TPTP (SYN007+1.005)*}
   820 lemma "p1 = (p2 = (p3 = (p4 = (p5 = (p1 = (p2 = (p3 = (p4 = p5))))))))"
   821 by blast
   822 
   823 text{*A manual resolution proof of problem 19.*}
   824 lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
   825 proof (rule ccontr, skolemize, make_clauses)
   826   fix x
   827   assume P: "\<And>U. \<not> P U \<Longrightarrow> False" 
   828      and Q: "\<And>U. Q U \<Longrightarrow> False"
   829      and PQ: "\<lbrakk>P x; \<not> Q x\<rbrakk> \<Longrightarrow> False"
   830   have cl4: "\<And>U. \<not> Q x \<Longrightarrow> False"
   831     by (rule P [binary 0 PQ 0])
   832   show "False"
   833     by (rule Q [binary 0 cl4 0])
   834 qed
   835 
   836 end