src/HOL/ex/Classical.thy
 author bulwahn Fri Oct 21 11:17:14 2011 +0200 (2011-10-21) changeset 45231 d85a2fdc586c parent 41959 b460124855b8 child 58889 5b7a9633cfa8 permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
1 (*  Title:      HOL/ex/Classical.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1994  University of Cambridge
4 *)
8 theory Classical imports Main begin
12 text{*The machine "griffon" mentioned below is a 2.5GHz Power Mac G5.*}
14 text{*Taken from @{text "FOL/Classical.thy"}. When porting examples from
15 first-order logic, beware of the precedence of @{text "="} versus @{text
16 "\<leftrightarrow>"}.*}
18 lemma "(P --> Q | R) --> (P-->Q) | (P-->R)"
19 by blast
21 text{*If and only if*}
23 lemma "(P=Q) = (Q = (P::bool))"
24 by blast
26 lemma "~ (P = (~P))"
27 by blast
30 text{*Sample problems from
31   F. J. Pelletier,
32   Seventy-Five Problems for Testing Automatic Theorem Provers,
33   J. Automated Reasoning 2 (1986), 191-216.
34   Errata, JAR 4 (1988), 236-236.
36 The hardest problems -- judging by experience with several theorem provers,
37 including matrix ones -- are 34 and 43.
38 *}
40 subsubsection{*Pelletier's examples*}
42 text{*1*}
43 lemma "(P-->Q)  =  (~Q --> ~P)"
44 by blast
46 text{*2*}
47 lemma "(~ ~ P) =  P"
48 by blast
50 text{*3*}
51 lemma "~(P-->Q) --> (Q-->P)"
52 by blast
54 text{*4*}
55 lemma "(~P-->Q)  =  (~Q --> P)"
56 by blast
58 text{*5*}
59 lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
60 by blast
62 text{*6*}
63 lemma "P | ~ P"
64 by blast
66 text{*7*}
67 lemma "P | ~ ~ ~ P"
68 by blast
70 text{*8.  Peirce's law*}
71 lemma "((P-->Q) --> P)  -->  P"
72 by blast
74 text{*9*}
75 lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
76 by blast
78 text{*10*}
79 lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"
80 by blast
82 text{*11.  Proved in each direction (incorrectly, says Pelletier!!)  *}
83 lemma "P=(P::bool)"
84 by blast
86 text{*12.  "Dijkstra's law"*}
87 lemma "((P = Q) = R) = (P = (Q = R))"
88 by blast
90 text{*13.  Distributive law*}
91 lemma "(P | (Q & R)) = ((P | Q) & (P | R))"
92 by blast
94 text{*14*}
95 lemma "(P = Q) = ((Q | ~P) & (~Q|P))"
96 by blast
98 text{*15*}
99 lemma "(P --> Q) = (~P | Q)"
100 by blast
102 text{*16*}
103 lemma "(P-->Q) | (Q-->P)"
104 by blast
106 text{*17*}
107 lemma "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))"
108 by blast
110 subsubsection{*Classical Logic: examples with quantifiers*}
112 lemma "(\<forall>x. P(x) & Q(x)) = ((\<forall>x. P(x)) & (\<forall>x. Q(x)))"
113 by blast
115 lemma "(\<exists>x. P-->Q(x))  =  (P --> (\<exists>x. Q(x)))"
116 by blast
118 lemma "(\<exists>x. P(x)-->Q) = ((\<forall>x. P(x)) --> Q)"
119 by blast
121 lemma "((\<forall>x. P(x)) | Q)  =  (\<forall>x. P(x) | Q)"
122 by blast
124 text{*From Wishnu Prasetya*}
125 lemma "(\<forall>s. q(s) --> r(s)) & ~r(s) & (\<forall>s. ~r(s) & ~q(s) --> p(t) | q(t))
126     --> p(t) | r(t)"
127 by blast
130 subsubsection{*Problems requiring quantifier duplication*}
132 text{*Theorem B of Peter Andrews, Theorem Proving via General Matings,
133   JACM 28 (1981).*}
134 lemma "(\<exists>x. \<forall>y. P(x) = P(y)) --> ((\<exists>x. P(x)) = (\<forall>y. P(y)))"
135 by blast
137 text{*Needs multiple instantiation of the quantifier.*}
138 lemma "(\<forall>x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))"
139 by blast
141 text{*Needs double instantiation of the quantifier*}
142 lemma "\<exists>x. P(x) --> P(a) & P(b)"
143 by blast
145 lemma "\<exists>z. P(z) --> (\<forall>x. P(x))"
146 by blast
148 lemma "\<exists>x. (\<exists>y. P(y)) --> P(x)"
149 by blast
151 subsubsection{*Hard examples with quantifiers*}
153 text{*Problem 18*}
154 lemma "\<exists>y. \<forall>x. P(y)-->P(x)"
155 by blast
157 text{*Problem 19*}
158 lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
159 by blast
161 text{*Problem 20*}
162 lemma "(\<forall>x y. \<exists>z. \<forall>w. (P(x)&Q(y)-->R(z)&S(w)))
163     --> (\<exists>x y. P(x) & Q(y)) --> (\<exists>z. R(z))"
164 by blast
166 text{*Problem 21*}
167 lemma "(\<exists>x. P-->Q(x)) & (\<exists>x. Q(x)-->P) --> (\<exists>x. P=Q(x))"
168 by blast
170 text{*Problem 22*}
171 lemma "(\<forall>x. P = Q(x))  -->  (P = (\<forall>x. Q(x)))"
172 by blast
174 text{*Problem 23*}
175 lemma "(\<forall>x. P | Q(x))  =  (P | (\<forall>x. Q(x)))"
176 by blast
178 text{*Problem 24*}
179 lemma "~(\<exists>x. S(x)&Q(x)) & (\<forall>x. P(x) --> Q(x)|R(x)) &
180      (~(\<exists>x. P(x)) --> (\<exists>x. Q(x))) & (\<forall>x. Q(x)|R(x) --> S(x))
181     --> (\<exists>x. P(x)&R(x))"
182 by blast
184 text{*Problem 25*}
185 lemma "(\<exists>x. P(x)) &
186         (\<forall>x. L(x) --> ~ (M(x) & R(x))) &
187         (\<forall>x. P(x) --> (M(x) & L(x))) &
188         ((\<forall>x. P(x)-->Q(x)) | (\<exists>x. P(x)&R(x)))
189     --> (\<exists>x. Q(x)&P(x))"
190 by blast
192 text{*Problem 26*}
193 lemma "((\<exists>x. p(x)) = (\<exists>x. q(x))) &
194       (\<forall>x. \<forall>y. p(x) & q(y) --> (r(x) = s(y)))
195   --> ((\<forall>x. p(x)-->r(x)) = (\<forall>x. q(x)-->s(x)))"
196 by blast
198 text{*Problem 27*}
199 lemma "(\<exists>x. P(x) & ~Q(x)) &
200               (\<forall>x. P(x) --> R(x)) &
201               (\<forall>x. M(x) & L(x) --> P(x)) &
202               ((\<exists>x. R(x) & ~ Q(x)) --> (\<forall>x. L(x) --> ~ R(x)))
203           --> (\<forall>x. M(x) --> ~L(x))"
204 by blast
206 text{*Problem 28.  AMENDED*}
207 lemma "(\<forall>x. P(x) --> (\<forall>x. Q(x))) &
208         ((\<forall>x. Q(x)|R(x)) --> (\<exists>x. Q(x)&S(x))) &
209         ((\<exists>x. S(x)) --> (\<forall>x. L(x) --> M(x)))
210     --> (\<forall>x. P(x) & L(x) --> M(x))"
211 by blast
213 text{*Problem 29.  Essentially the same as Principia Mathematica *11.71*}
214 lemma "(\<exists>x. F(x)) & (\<exists>y. G(y))
215     --> ( ((\<forall>x. F(x)-->H(x)) & (\<forall>y. G(y)-->J(y)))  =
216           (\<forall>x y. F(x) & G(y) --> H(x) & J(y)))"
217 by blast
219 text{*Problem 30*}
220 lemma "(\<forall>x. P(x) | Q(x) --> ~ R(x)) &
221         (\<forall>x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
222     --> (\<forall>x. S(x))"
223 by blast
225 text{*Problem 31*}
226 lemma "~(\<exists>x. P(x) & (Q(x) | R(x))) &
227         (\<exists>x. L(x) & P(x)) &
228         (\<forall>x. ~ R(x) --> M(x))
229     --> (\<exists>x. L(x) & M(x))"
230 by blast
232 text{*Problem 32*}
233 lemma "(\<forall>x. P(x) & (Q(x)|R(x))-->S(x)) &
234         (\<forall>x. S(x) & R(x) --> L(x)) &
235         (\<forall>x. M(x) --> R(x))
236     --> (\<forall>x. P(x) & M(x) --> L(x))"
237 by blast
239 text{*Problem 33*}
240 lemma "(\<forall>x. P(a) & (P(x)-->P(b))-->P(c))  =
241      (\<forall>x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
242 by blast
244 text{*Problem 34  AMENDED (TWICE!!)*}
245 text{*Andrews's challenge*}
246 lemma "((\<exists>x. \<forall>y. p(x) = p(y))  =
247                ((\<exists>x. q(x)) = (\<forall>y. p(y))))   =
248               ((\<exists>x. \<forall>y. q(x) = q(y))  =
249                ((\<exists>x. p(x)) = (\<forall>y. q(y))))"
250 by blast
252 text{*Problem 35*}
253 lemma "\<exists>x y. P x y -->  (\<forall>u v. P u v)"
254 by blast
256 text{*Problem 36*}
257 lemma "(\<forall>x. \<exists>y. J x y) &
258         (\<forall>x. \<exists>y. G x y) &
259         (\<forall>x y. J x y | G x y -->
260         (\<forall>z. J y z | G y z --> H x z))
261     --> (\<forall>x. \<exists>y. H x y)"
262 by blast
264 text{*Problem 37*}
265 lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
266            (P x z -->P y w) & P y z & (P y w --> (\<exists>u. Q u w))) &
267         (\<forall>x z. ~(P x z) --> (\<exists>y. Q y z)) &
268         ((\<exists>x y. Q x y) --> (\<forall>x. R x x))
269     --> (\<forall>x. \<exists>y. R x y)"
270 by blast
272 text{*Problem 38*}
273 lemma "(\<forall>x. p(a) & (p(x) --> (\<exists>y. p(y) & r x y)) -->
274            (\<exists>z. \<exists>w. p(z) & r x w & r w z))  =
275      (\<forall>x. (~p(a) | p(x) | (\<exists>z. \<exists>w. p(z) & r x w & r w z)) &
276            (~p(a) | ~(\<exists>y. p(y) & r x y) |
277             (\<exists>z. \<exists>w. p(z) & r x w & r w z)))"
278 by blast (*beats fast!*)
280 text{*Problem 39*}
281 lemma "~ (\<exists>x. \<forall>y. F y x = (~ F y y))"
282 by blast
284 text{*Problem 40.  AMENDED*}
285 lemma "(\<exists>y. \<forall>x. F x y = F x x)
286         -->  ~ (\<forall>x. \<exists>y. \<forall>z. F z y = (~ F z x))"
287 by blast
289 text{*Problem 41*}
290 lemma "(\<forall>z. \<exists>y. \<forall>x. f x y = (f x z & ~ f x x))
291                --> ~ (\<exists>z. \<forall>x. f x z)"
292 by blast
294 text{*Problem 42*}
295 lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
296 by blast
298 text{*Problem 43!!*}
299 lemma "(\<forall>x::'a. \<forall>y::'a. q x y = (\<forall>z. p z x = (p z y::bool)))
300   --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
301 by blast
303 text{*Problem 44*}
304 lemma "(\<forall>x. f(x) -->
305               (\<exists>y. g(y) & h x y & (\<exists>y. g(y) & ~ h x y)))  &
306               (\<exists>x. j(x) & (\<forall>y. g(y) --> h x y))
307               --> (\<exists>x. j(x) & ~f(x))"
308 by blast
310 text{*Problem 45*}
311 lemma "(\<forall>x. f(x) & (\<forall>y. g(y) & h x y --> j x y)
312                       --> (\<forall>y. g(y) & h x y --> k(y))) &
313      ~ (\<exists>y. l(y) & k(y)) &
314      (\<exists>x. f(x) & (\<forall>y. h x y --> l(y))
315                 & (\<forall>y. g(y) & h x y --> j x y))
316       --> (\<exists>x. f(x) & ~ (\<exists>y. g(y) & h x y))"
317 by blast
320 subsubsection{*Problems (mainly) involving equality or functions*}
322 text{*Problem 48*}
323 lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
324 by blast
326 text{*Problem 49  NOT PROVED AUTOMATICALLY.
327      Hard because it involves substitution for Vars
328   the type constraint ensures that x,y,z have the same type as a,b,u. *}
329 lemma "(\<exists>x y::'a. \<forall>z. z=x | z=y) & P(a) & P(b) & (~a=b)
330                 --> (\<forall>u::'a. P(u))"
331 by metis
333 text{*Problem 50.  (What has this to do with equality?) *}
334 lemma "(\<forall>x. P a x | (\<forall>y. P x y)) --> (\<exists>x. \<forall>y. P x y)"
335 by blast
337 text{*Problem 51*}
338 lemma "(\<exists>z w. \<forall>x y. P x y = (x=z & y=w)) -->
339      (\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P x y = (y=w)) = (x=z))"
340 by blast
342 text{*Problem 52. Almost the same as 51. *}
343 lemma "(\<exists>z w. \<forall>x y. P x y = (x=z & y=w)) -->
344      (\<exists>w. \<forall>y. \<exists>z. (\<forall>x. P x y = (x=z)) = (y=w))"
345 by blast
347 text{*Problem 55*}
349 text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
350   fast DISCOVERS who killed Agatha. *}
351 schematic_lemma "lives(agatha) & lives(butler) & lives(charles) &
352    (killed agatha agatha | killed butler agatha | killed charles agatha) &
353    (\<forall>x y. killed x y --> hates x y & ~richer x y) &
354    (\<forall>x. hates agatha x --> ~hates charles x) &
355    (hates agatha agatha & hates agatha charles) &
356    (\<forall>x. lives(x) & ~richer x agatha --> hates butler x) &
357    (\<forall>x. hates agatha x --> hates butler x) &
358    (\<forall>x. ~hates x agatha | ~hates x butler | ~hates x charles) -->
359     killed ?who agatha"
360 by fast
362 text{*Problem 56*}
363 lemma "(\<forall>x. (\<exists>y. P(y) & x=f(y)) --> P(x)) = (\<forall>x. P(x) --> P(f(x)))"
364 by blast
366 text{*Problem 57*}
367 lemma "P (f a b) (f b c) & P (f b c) (f a c) &
368      (\<forall>x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)"
369 by blast
371 text{*Problem 58  NOT PROVED AUTOMATICALLY*}
372 lemma "(\<forall>x y. f(x)=g(y)) --> (\<forall>x y. f(f(x))=f(g(y)))"
373 by (fast intro: arg_cong [of concl: f])
375 text{*Problem 59*}
376 lemma "(\<forall>x. P(x) = (~P(f(x)))) --> (\<exists>x. P(x) & ~P(f(x)))"
377 by blast
379 text{*Problem 60*}
380 lemma "\<forall>x. P x (f x) = (\<exists>y. (\<forall>z. P z y --> P z (f x)) & P x y)"
381 by blast
383 text{*Problem 62 as corrected in JAR 18 (1997), page 135*}
384 lemma "(\<forall>x. p a & (p x --> p(f x)) --> p(f(f x)))  =
385       (\<forall>x. (~ p a | p x | p(f(f x))) &
386               (~ p a | ~ p(f x) | p(f(f x))))"
387 by blast
389 text{*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
390   fast indeed copes!*}
391 lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
392        (\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
393        (\<forall>x. K(x) --> ~G(x))  -->  (\<exists>x. K(x) & J(x))"
394 by fast
396 text{*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
397   It does seem obvious!*}
398 lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
399        (\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y)))  &
400        (\<forall>x. K(x) --> ~G(x))   -->   (\<exists>x. K(x) --> ~G(x))"
401 by fast
403 text{*Attributed to Lewis Carroll by S. G. Pulman.  The first or last
404 assumption can be deleted.*}
405 lemma "(\<forall>x. honest(x) & industrious(x) --> healthy(x)) &
406       ~ (\<exists>x. grocer(x) & healthy(x)) &
407       (\<forall>x. industrious(x) & grocer(x) --> honest(x)) &
408       (\<forall>x. cyclist(x) --> industrious(x)) &
409       (\<forall>x. ~healthy(x) & cyclist(x) --> ~honest(x))
410       --> (\<forall>x. grocer(x) --> ~cyclist(x))"
411 by blast
413 lemma "(\<forall>x y. R(x,y) | R(y,x)) &
414        (\<forall>x y. S(x,y) & S(y,x) --> x=y) &
415        (\<forall>x y. R(x,y) --> S(x,y))    -->   (\<forall>x y. S(x,y) --> R(x,y))"
416 by blast
419 subsection{*Model Elimination Prover*}
422 text{*Trying out meson with arguments*}
423 lemma "x < y & y < z --> ~ (z < (x::nat))"
424 by (meson order_less_irrefl order_less_trans)
426 text{*The "small example" from Bezem, Hendriks and de Nivelle,
427 Automatic Proof Construction in Type Theory Using Resolution,
428 JAR 29: 3-4 (2002), pages 253-275 *}
429 lemma "(\<forall>x y z. R(x,y) & R(y,z) --> R(x,z)) &
430        (\<forall>x. \<exists>y. R(x,y)) -->
431        ~ (\<forall>x. P x = (\<forall>y. R(x,y) --> ~ P y))"
432 by (tactic{*Meson.safe_best_meson_tac @{context} 1*})
433     --{*In contrast, @{text meson} is SLOW: 7.6s on griffon*}
436 subsubsection{*Pelletier's examples*}
437 text{*1*}
438 lemma "(P --> Q)  =  (~Q --> ~P)"
439 by blast
441 text{*2*}
442 lemma "(~ ~ P) =  P"
443 by blast
445 text{*3*}
446 lemma "~(P-->Q) --> (Q-->P)"
447 by blast
449 text{*4*}
450 lemma "(~P-->Q)  =  (~Q --> P)"
451 by blast
453 text{*5*}
454 lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
455 by blast
457 text{*6*}
458 lemma "P | ~ P"
459 by blast
461 text{*7*}
462 lemma "P | ~ ~ ~ P"
463 by blast
465 text{*8.  Peirce's law*}
466 lemma "((P-->Q) --> P)  -->  P"
467 by blast
469 text{*9*}
470 lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
471 by blast
473 text{*10*}
474 lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"
475 by blast
477 text{*11.  Proved in each direction (incorrectly, says Pelletier!!)  *}
478 lemma "P=(P::bool)"
479 by blast
481 text{*12.  "Dijkstra's law"*}
482 lemma "((P = Q) = R) = (P = (Q = R))"
483 by blast
485 text{*13.  Distributive law*}
486 lemma "(P | (Q & R)) = ((P | Q) & (P | R))"
487 by blast
489 text{*14*}
490 lemma "(P = Q) = ((Q | ~P) & (~Q|P))"
491 by blast
493 text{*15*}
494 lemma "(P --> Q) = (~P | Q)"
495 by blast
497 text{*16*}
498 lemma "(P-->Q) | (Q-->P)"
499 by blast
501 text{*17*}
502 lemma "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))"
503 by blast
505 subsubsection{*Classical Logic: examples with quantifiers*}
507 lemma "(\<forall>x. P x & Q x) = ((\<forall>x. P x) & (\<forall>x. Q x))"
508 by blast
510 lemma "(\<exists>x. P --> Q x)  =  (P --> (\<exists>x. Q x))"
511 by blast
513 lemma "(\<exists>x. P x --> Q) = ((\<forall>x. P x) --> Q)"
514 by blast
516 lemma "((\<forall>x. P x) | Q)  =  (\<forall>x. P x | Q)"
517 by blast
519 lemma "(\<forall>x. P x --> P(f x))  &  P d --> P(f(f(f d)))"
520 by blast
522 text{*Needs double instantiation of EXISTS*}
523 lemma "\<exists>x. P x --> P a & P b"
524 by blast
526 lemma "\<exists>z. P z --> (\<forall>x. P x)"
527 by blast
529 text{*From a paper by Claire Quigley*}
530 lemma "\<exists>y. ((P c & Q y) | (\<exists>z. ~ Q z)) | (\<exists>x. ~ P x & Q d)"
531 by fast
533 subsubsection{*Hard examples with quantifiers*}
535 text{*Problem 18*}
536 lemma "\<exists>y. \<forall>x. P y --> P x"
537 by blast
539 text{*Problem 19*}
540 lemma "\<exists>x. \<forall>y z. (P y --> Q z) --> (P x --> Q x)"
541 by blast
543 text{*Problem 20*}
544 lemma "(\<forall>x y. \<exists>z. \<forall>w. (P x & Q y --> R z & S w))
545     --> (\<exists>x y. P x & Q y) --> (\<exists>z. R z)"
546 by blast
548 text{*Problem 21*}
549 lemma "(\<exists>x. P --> Q x) & (\<exists>x. Q x --> P) --> (\<exists>x. P=Q x)"
550 by blast
552 text{*Problem 22*}
553 lemma "(\<forall>x. P = Q x)  -->  (P = (\<forall>x. Q x))"
554 by blast
556 text{*Problem 23*}
557 lemma "(\<forall>x. P | Q x)  =  (P | (\<forall>x. Q x))"
558 by blast
560 text{*Problem 24*}  (*The first goal clause is useless*)
561 lemma "~(\<exists>x. S x & Q x) & (\<forall>x. P x --> Q x | R x) &
562       (~(\<exists>x. P x) --> (\<exists>x. Q x)) & (\<forall>x. Q x | R x --> S x)
563     --> (\<exists>x. P x & R x)"
564 by blast
566 text{*Problem 25*}
567 lemma "(\<exists>x. P x) &
568       (\<forall>x. L x --> ~ (M x & R x)) &
569       (\<forall>x. P x --> (M x & L x)) &
570       ((\<forall>x. P x --> Q x) | (\<exists>x. P x & R x))
571     --> (\<exists>x. Q x & P x)"
572 by blast
574 text{*Problem 26; has 24 Horn clauses*}
575 lemma "((\<exists>x. p x) = (\<exists>x. q x)) &
576       (\<forall>x. \<forall>y. p x & q y --> (r x = s y))
577   --> ((\<forall>x. p x --> r x) = (\<forall>x. q x --> s x))"
578 by blast
580 text{*Problem 27; has 13 Horn clauses*}
581 lemma "(\<exists>x. P x & ~Q x) &
582       (\<forall>x. P x --> R x) &
583       (\<forall>x. M x & L x --> P x) &
584       ((\<exists>x. R x & ~ Q x) --> (\<forall>x. L x --> ~ R x))
585       --> (\<forall>x. M x --> ~L x)"
586 by blast
588 text{*Problem 28.  AMENDED; has 14 Horn clauses*}
589 lemma "(\<forall>x. P x --> (\<forall>x. Q x)) &
590       ((\<forall>x. Q x | R x) --> (\<exists>x. Q x & S x)) &
591       ((\<exists>x. S x) --> (\<forall>x. L x --> M x))
592     --> (\<forall>x. P x & L x --> M x)"
593 by blast
595 text{*Problem 29.  Essentially the same as Principia Mathematica *11.71.
596       62 Horn clauses*}
597 lemma "(\<exists>x. F x) & (\<exists>y. G y)
598     --> ( ((\<forall>x. F x --> H x) & (\<forall>y. G y --> J y))  =
599           (\<forall>x y. F x & G y --> H x & J y))"
600 by blast
603 text{*Problem 30*}
604 lemma "(\<forall>x. P x | Q x --> ~ R x) & (\<forall>x. (Q x --> ~ S x) --> P x & R x)
605        --> (\<forall>x. S x)"
606 by blast
608 text{*Problem 31; has 10 Horn clauses; first negative clauses is useless*}
609 lemma "~(\<exists>x. P x & (Q x | R x)) &
610       (\<exists>x. L x & P x) &
611       (\<forall>x. ~ R x --> M x)
612     --> (\<exists>x. L x & M x)"
613 by blast
615 text{*Problem 32*}
616 lemma "(\<forall>x. P x & (Q x | R x)-->S x) &
617       (\<forall>x. S x & R x --> L x) &
618       (\<forall>x. M x --> R x)
619     --> (\<forall>x. P x & M x --> L x)"
620 by blast
622 text{*Problem 33; has 55 Horn clauses*}
623 lemma "(\<forall>x. P a & (P x --> P b)-->P c)  =
624       (\<forall>x. (~P a | P x | P c) & (~P a | ~P b | P c))"
625 by blast
627 text{*Problem 34: Andrews's challenge has 924 Horn clauses*}
628 lemma "((\<exists>x. \<forall>y. p x = p y)  = ((\<exists>x. q x) = (\<forall>y. p y)))     =
629       ((\<exists>x. \<forall>y. q x = q y)  = ((\<exists>x. p x) = (\<forall>y. q y)))"
630 by blast
632 text{*Problem 35*}
633 lemma "\<exists>x y. P x y -->  (\<forall>u v. P u v)"
634 by blast
636 text{*Problem 36; has 15 Horn clauses*}
637 lemma "(\<forall>x. \<exists>y. J x y) & (\<forall>x. \<exists>y. G x y) &
638        (\<forall>x y. J x y | G x y --> (\<forall>z. J y z | G y z --> H x z))
639        --> (\<forall>x. \<exists>y. H x y)"
640 by blast
642 text{*Problem 37; has 10 Horn clauses*}
643 lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
644            (P x z --> P y w) & P y z & (P y w --> (\<exists>u. Q u w))) &
645       (\<forall>x z. ~P x z --> (\<exists>y. Q y z)) &
646       ((\<exists>x y. Q x y) --> (\<forall>x. R x x))
647     --> (\<forall>x. \<exists>y. R x y)"
648 by blast --{*causes unification tracing messages*}
651 text{*Problem 38*}  text{*Quite hard: 422 Horn clauses!!*}
652 lemma "(\<forall>x. p a & (p x --> (\<exists>y. p y & r x y)) -->
653            (\<exists>z. \<exists>w. p z & r x w & r w z))  =
654       (\<forall>x. (~p a | p x | (\<exists>z. \<exists>w. p z & r x w & r w z)) &
655             (~p a | ~(\<exists>y. p y & r x y) |
656              (\<exists>z. \<exists>w. p z & r x w & r w z)))"
657 by blast
659 text{*Problem 39*}
660 lemma "~ (\<exists>x. \<forall>y. F y x = (~F y y))"
661 by blast
663 text{*Problem 40.  AMENDED*}
664 lemma "(\<exists>y. \<forall>x. F x y = F x x)
665       -->  ~ (\<forall>x. \<exists>y. \<forall>z. F z y = (~F z x))"
666 by blast
668 text{*Problem 41*}
669 lemma "(\<forall>z. (\<exists>y. (\<forall>x. f x y = (f x z & ~ f x x))))
670       --> ~ (\<exists>z. \<forall>x. f x z)"
671 by blast
673 text{*Problem 42*}
674 lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
675 by blast
677 text{*Problem 43  NOW PROVED AUTOMATICALLY!!*}
678 lemma "(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool)))
679       --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
680 by blast
682 text{*Problem 44: 13 Horn clauses; 7-step proof*}
683 lemma "(\<forall>x. f x --> (\<exists>y. g y & h x y & (\<exists>y. g y & ~ h x y)))  &
684        (\<exists>x. j x & (\<forall>y. g y --> h x y))
685        --> (\<exists>x. j x & ~f x)"
686 by blast
688 text{*Problem 45; has 27 Horn clauses; 54-step proof*}
689 lemma "(\<forall>x. f x & (\<forall>y. g y & h x y --> j x y)
690             --> (\<forall>y. g y & h x y --> k y)) &
691       ~ (\<exists>y. l y & k y) &
692       (\<exists>x. f x & (\<forall>y. h x y --> l y)
693                 & (\<forall>y. g y & h x y --> j x y))
694       --> (\<exists>x. f x & ~ (\<exists>y. g y & h x y))"
695 by blast
697 text{*Problem 46; has 26 Horn clauses; 21-step proof*}
698 lemma "(\<forall>x. f x & (\<forall>y. f y & h y x --> g y) --> g x) &
699        ((\<exists>x. f x & ~g x) -->
700        (\<exists>x. f x & ~g x & (\<forall>y. f y & ~g y --> j x y))) &
701        (\<forall>x y. f x & f y & h x y --> ~j y x)
702        --> (\<forall>x. f x --> g x)"
703 by blast
705 text{*Problem 47.  Schubert's Steamroller.
706       26 clauses; 63 Horn clauses.
707       87094 inferences so far.  Searching to depth 36*}
708 lemma "(\<forall>x. wolf x \<longrightarrow> animal x) & (\<exists>x. wolf x) &
709        (\<forall>x. fox x \<longrightarrow> animal x) & (\<exists>x. fox x) &
710        (\<forall>x. bird x \<longrightarrow> animal x) & (\<exists>x. bird x) &
711        (\<forall>x. caterpillar x \<longrightarrow> animal x) & (\<exists>x. caterpillar x) &
712        (\<forall>x. snail x \<longrightarrow> animal x) & (\<exists>x. snail x) &
713        (\<forall>x. grain x \<longrightarrow> plant x) & (\<exists>x. grain x) &
714        (\<forall>x. animal x \<longrightarrow>
715              ((\<forall>y. plant y \<longrightarrow> eats x y)  \<or>
716               (\<forall>y. animal y & smaller_than y x &
717                     (\<exists>z. plant z & eats y z) \<longrightarrow> eats x y))) &
718        (\<forall>x y. bird y & (snail x \<or> caterpillar x) \<longrightarrow> smaller_than x y) &
719        (\<forall>x y. bird x & fox y \<longrightarrow> smaller_than x y) &
720        (\<forall>x y. fox x & wolf y \<longrightarrow> smaller_than x y) &
721        (\<forall>x y. wolf x & (fox y \<or> grain y) \<longrightarrow> ~eats x y) &
722        (\<forall>x y. bird x & caterpillar y \<longrightarrow> eats x y) &
723        (\<forall>x y. bird x & snail y \<longrightarrow> ~eats x y) &
724        (\<forall>x. (caterpillar x \<or> snail x) \<longrightarrow> (\<exists>y. plant y & eats x y))
725        \<longrightarrow> (\<exists>x y. animal x & animal y & (\<exists>z. grain z & eats y z & eats x y))"
726 by (tactic{*Meson.safe_best_meson_tac @{context} 1*})
727     --{*Nearly twice as fast as @{text meson},
728         which performs iterative deepening rather than best-first search*}
730 text{*The Los problem. Circulated by John Harrison*}
731 lemma "(\<forall>x y z. P x y & P y z --> P x z) &
732        (\<forall>x y z. Q x y & Q y z --> Q x z) &
733        (\<forall>x y. P x y --> P y x) &
734        (\<forall>x y. P x y | Q x y)
735        --> (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
736 by meson
738 text{*A similar example, suggested by Johannes Schumann and
739  credited to Pelletier*}
740 lemma "(\<forall>x y z. P x y --> P y z --> P x z) -->
741        (\<forall>x y z. Q x y --> Q y z --> Q x z) -->
742        (\<forall>x y. Q x y --> Q y x) -->  (\<forall>x y. P x y | Q x y) -->
743        (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
744 by meson
746 text{*Problem 50.  What has this to do with equality?*}
747 lemma "(\<forall>x. P a x | (\<forall>y. P x y)) --> (\<exists>x. \<forall>y. P x y)"
748 by blast
750 text{*Problem 54: NOT PROVED*}
751 lemma "(\<forall>y::'a. \<exists>z. \<forall>x. F x z = (x=y)) -->
752       ~ (\<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))))"
753 oops
756 text{*Problem 55*}
758 text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
759   @{text meson} cannot report who killed Agatha. *}
760 lemma "lives agatha & lives butler & lives charles &
761        (killed agatha agatha | killed butler agatha | killed charles agatha) &
762        (\<forall>x y. killed x y --> hates x y & ~richer x y) &
763        (\<forall>x. hates agatha x --> ~hates charles x) &
764        (hates agatha agatha & hates agatha charles) &
765        (\<forall>x. lives x & ~richer x agatha --> hates butler x) &
766        (\<forall>x. hates agatha x --> hates butler x) &
767        (\<forall>x. ~hates x agatha | ~hates x butler | ~hates x charles) -->
768        (\<exists>x. killed x agatha)"
769 by meson
771 text{*Problem 57*}
772 lemma "P (f a b) (f b c) & P (f b c) (f a c) &
773       (\<forall>x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)"
774 by blast
776 text{*Problem 58: Challenge found on info-hol *}
777 lemma "\<forall>P Q R x. \<exists>v w. \<forall>y z. P x & Q y --> (P v | R w) & (R z --> Q v)"
778 by blast
780 text{*Problem 59*}
781 lemma "(\<forall>x. P x = (~P(f x))) --> (\<exists>x. P x & ~P(f x))"
782 by blast
784 text{*Problem 60*}
785 lemma "\<forall>x. P x (f x) = (\<exists>y. (\<forall>z. P z y --> P z (f x)) & P x y)"
786 by blast
788 text{*Problem 62 as corrected in JAR 18 (1997), page 135*}
789 lemma "(\<forall>x. p a & (p x --> p(f x)) --> p(f(f x)))  =
790        (\<forall>x. (~ p a | p x | p(f(f x))) &
791             (~ p a | ~ p(f x) | p(f(f x))))"
792 by blast
794 text{** Charles Morgan's problems **}
796 lemma
797   assumes a: "\<forall>x y.  T(i x(i y x))"
798       and b: "\<forall>x y z. T(i (i x (i y z)) (i (i x y) (i x z)))"
799       and c: "\<forall>x y.   T(i (i (n x) (n y)) (i y x))"
800       and c': "\<forall>x y.   T(i (i y x) (i (n x) (n y)))"
801       and d: "\<forall>x y.   T(i x y) & T x --> T y"
802  shows True
803 proof -
804   from a b d have "\<forall>x. T(i x x)" by blast
805   from a b c d have "\<forall>x. T(i x (n(n x)))" --{*Problem 66*}
806     by metis
807   from a b c d have "\<forall>x. T(i (n(n x)) x)" --{*Problem 67*}
808     by meson
809       --{*4.9s on griffon. 51061 inferences, depth 21 *}
810   from a b c' d have "\<forall>x. T(i x (n(n x)))"
811       --{*Problem 68: not proved.  Listed as satisfiable in TPTP (LCL078-1)*}
812 oops
814 text{*Problem 71, as found in TPTP (SYN007+1.005)*}
815 lemma "p1 = (p2 = (p3 = (p4 = (p5 = (p1 = (p2 = (p3 = (p4 = p5))))))))"
816 by blast
818 end