author | paulson |
Thu, 14 Dec 1995 12:49:32 +0100 | |
changeset 1404 | 57c3f6d2e692 |
parent 732 | 584b3475e859 |
child 1459 | d12da312eff4 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: FOL/ex/cla |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
4 |
Copyright 1994 University of Cambridge |
0 | 5 |
|
6 |
Classical First-Order Logic |
|
7 |
*) |
|
8 |
||
9 |
writeln"File FOL/ex/cla."; |
|
10 |
||
11 |
open Cla; (*in case structure Int is open!*) |
|
12 |
||
13 |
goal FOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)"; |
|
14 |
by (fast_tac FOL_cs 1); |
|
15 |
result(); |
|
16 |
||
17 |
(*If and only if*) |
|
18 |
||
19 |
goal FOL.thy "(P<->Q) <-> (Q<->P)"; |
|
20 |
by (fast_tac FOL_cs 1); |
|
21 |
result(); |
|
22 |
||
23 |
goal FOL.thy "~ (P <-> ~P)"; |
|
24 |
by (fast_tac FOL_cs 1); |
|
25 |
result(); |
|
26 |
||
27 |
||
28 |
(*Sample problems from |
|
29 |
F. J. Pelletier, |
|
30 |
Seventy-Five Problems for Testing Automatic Theorem Provers, |
|
31 |
J. Automated Reasoning 2 (1986), 191-216. |
|
32 |
Errata, JAR 4 (1988), 236-236. |
|
33 |
||
34 |
The hardest problems -- judging by experience with several theorem provers, |
|
35 |
including matrix ones -- are 34 and 43. |
|
36 |
*) |
|
37 |
||
38 |
writeln"Pelletier's examples"; |
|
39 |
(*1*) |
|
40 |
goal FOL.thy "(P-->Q) <-> (~Q --> ~P)"; |
|
41 |
by (fast_tac FOL_cs 1); |
|
42 |
result(); |
|
43 |
||
44 |
(*2*) |
|
45 |
goal FOL.thy "~ ~ P <-> P"; |
|
46 |
by (fast_tac FOL_cs 1); |
|
47 |
result(); |
|
48 |
||
49 |
(*3*) |
|
50 |
goal FOL.thy "~(P-->Q) --> (Q-->P)"; |
|
51 |
by (fast_tac FOL_cs 1); |
|
52 |
result(); |
|
53 |
||
54 |
(*4*) |
|
55 |
goal FOL.thy "(~P-->Q) <-> (~Q --> P)"; |
|
56 |
by (fast_tac FOL_cs 1); |
|
57 |
result(); |
|
58 |
||
59 |
(*5*) |
|
60 |
goal FOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))"; |
|
61 |
by (fast_tac FOL_cs 1); |
|
62 |
result(); |
|
63 |
||
64 |
(*6*) |
|
65 |
goal FOL.thy "P | ~ P"; |
|
66 |
by (fast_tac FOL_cs 1); |
|
67 |
result(); |
|
68 |
||
69 |
(*7*) |
|
70 |
goal FOL.thy "P | ~ ~ ~ P"; |
|
71 |
by (fast_tac FOL_cs 1); |
|
72 |
result(); |
|
73 |
||
74 |
(*8. Peirce's law*) |
|
75 |
goal FOL.thy "((P-->Q) --> P) --> P"; |
|
76 |
by (fast_tac FOL_cs 1); |
|
77 |
result(); |
|
78 |
||
79 |
(*9*) |
|
80 |
goal FOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; |
|
81 |
by (fast_tac FOL_cs 1); |
|
82 |
result(); |
|
83 |
||
84 |
(*10*) |
|
85 |
goal FOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"; |
|
86 |
by (fast_tac FOL_cs 1); |
|
87 |
result(); |
|
88 |
||
89 |
(*11. Proved in each direction (incorrectly, says Pelletier!!) *) |
|
90 |
goal FOL.thy "P<->P"; |
|
91 |
by (fast_tac FOL_cs 1); |
|
92 |
result(); |
|
93 |
||
94 |
(*12. "Dijkstra's law"*) |
|
95 |
goal FOL.thy "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"; |
|
96 |
by (fast_tac FOL_cs 1); |
|
97 |
result(); |
|
98 |
||
99 |
(*13. Distributive law*) |
|
100 |
goal FOL.thy "P | (Q & R) <-> (P | Q) & (P | R)"; |
|
101 |
by (fast_tac FOL_cs 1); |
|
102 |
result(); |
|
103 |
||
104 |
(*14*) |
|
105 |
goal FOL.thy "(P <-> Q) <-> ((Q | ~P) & (~Q|P))"; |
|
106 |
by (fast_tac FOL_cs 1); |
|
107 |
result(); |
|
108 |
||
109 |
(*15*) |
|
110 |
goal FOL.thy "(P --> Q) <-> (~P | Q)"; |
|
111 |
by (fast_tac FOL_cs 1); |
|
112 |
result(); |
|
113 |
||
114 |
(*16*) |
|
115 |
goal FOL.thy "(P-->Q) | (Q-->P)"; |
|
116 |
by (fast_tac FOL_cs 1); |
|
117 |
result(); |
|
118 |
||
119 |
(*17*) |
|
120 |
goal FOL.thy "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"; |
|
121 |
by (fast_tac FOL_cs 1); |
|
122 |
result(); |
|
123 |
||
124 |
writeln"Classical Logic: examples with quantifiers"; |
|
125 |
||
126 |
goal FOL.thy "(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; |
|
127 |
by (fast_tac FOL_cs 1); |
|
128 |
result(); |
|
129 |
||
130 |
goal FOL.thy "(EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))"; |
|
131 |
by (fast_tac FOL_cs 1); |
|
132 |
result(); |
|
133 |
||
134 |
goal FOL.thy "(EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q"; |
|
135 |
by (fast_tac FOL_cs 1); |
|
136 |
result(); |
|
137 |
||
138 |
goal FOL.thy "(ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)"; |
|
139 |
by (fast_tac FOL_cs 1); |
|
140 |
result(); |
|
141 |
||
13
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
142 |
(*Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux, |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
143 |
JAR 10 (265-281), 1993. Proof is trivial!*) |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
144 |
goal FOL.thy |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
145 |
"~ ((EX x.~P(x)) & ((EX x.P(x)) | (EX x.P(x) & Q(x))) & ~ (EX x.P(x)))"; |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
146 |
by (fast_tac FOL_cs 1); |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
147 |
result(); |
b73f7e42135e
Added example from Avron: Gentzen-Type Systems, Resolution and Tableaux, JAR 10
lcp
parents:
0
diff
changeset
|
148 |
|
0 | 149 |
writeln"Problems requiring quantifier duplication"; |
150 |
||
151 |
(*Needs multiple instantiation of ALL.*) |
|
152 |
goal FOL.thy "(ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; |
|
732 | 153 |
by (deepen_tac FOL_cs 0 1); |
0 | 154 |
result(); |
155 |
||
156 |
(*Needs double instantiation of the quantifier*) |
|
157 |
goal FOL.thy "EX x. P(x) --> P(a) & P(b)"; |
|
732 | 158 |
by (deepen_tac FOL_cs 0 1); |
0 | 159 |
result(); |
160 |
||
161 |
goal FOL.thy "EX z. P(z) --> (ALL x. P(x))"; |
|
732 | 162 |
by (deepen_tac FOL_cs 0 1); |
0 | 163 |
result(); |
164 |
||
492 | 165 |
goal FOL.thy "EX x. (EX y. P(y)) --> P(x)"; |
732 | 166 |
by (deepen_tac FOL_cs 0 1); |
492 | 167 |
result(); |
168 |
||
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
169 |
(*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*) |
0 | 170 |
goal FOL.thy "EX x x'. ALL y. EX z z'. \ |
171 |
\ (~P(y,y) | P(x,x) | ~S(z,x)) & \ |
|
172 |
\ (S(x,y) | ~S(y,z) | Q(z',z')) & \ |
|
173 |
\ (Q(x',y) | ~Q(y,z') | S(x',x'))"; |
|
174 |
||
175 |
writeln"Hard examples with quantifiers"; |
|
176 |
||
177 |
writeln"Problem 18"; |
|
178 |
goal FOL.thy "EX y. ALL x. P(y)-->P(x)"; |
|
732 | 179 |
by (deepen_tac FOL_cs 0 1); |
0 | 180 |
result(); |
181 |
||
182 |
writeln"Problem 19"; |
|
183 |
goal FOL.thy "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; |
|
732 | 184 |
by (deepen_tac FOL_cs 0 1); |
0 | 185 |
result(); |
186 |
||
187 |
writeln"Problem 20"; |
|
188 |
goal FOL.thy "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \ |
|
189 |
\ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"; |
|
190 |
by (fast_tac FOL_cs 1); |
|
191 |
result(); |
|
192 |
||
193 |
writeln"Problem 21"; |
|
194 |
goal FOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; |
|
732 | 195 |
by (deepen_tac FOL_cs 0 1); |
0 | 196 |
result(); |
197 |
||
198 |
writeln"Problem 22"; |
|
199 |
goal FOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; |
|
200 |
by (fast_tac FOL_cs 1); |
|
201 |
result(); |
|
202 |
||
203 |
writeln"Problem 23"; |
|
204 |
goal FOL.thy "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; |
|
205 |
by (best_tac FOL_cs 1); |
|
206 |
result(); |
|
207 |
||
208 |
writeln"Problem 24"; |
|
209 |
goal FOL.thy "~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \ |
|
210 |
\ ~(EX x.P(x)) --> (EX x.Q(x)) & (ALL x. Q(x)|R(x) --> S(x)) \ |
|
211 |
\ --> (EX x. P(x)&R(x))"; |
|
212 |
by (fast_tac FOL_cs 1); |
|
213 |
result(); |
|
214 |
||
215 |
writeln"Problem 25"; |
|
216 |
goal FOL.thy "(EX x. P(x)) & \ |
|
217 |
\ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ |
|
218 |
\ (ALL x. P(x) --> (M(x) & L(x))) & \ |
|
219 |
\ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ |
|
220 |
\ --> (EX x. Q(x)&P(x))"; |
|
221 |
by (best_tac FOL_cs 1); |
|
222 |
result(); |
|
223 |
||
224 |
writeln"Problem 26"; |
|
225 |
goal FOL.thy "((EX x. p(x)) <-> (EX x. q(x))) & \ |
|
226 |
\ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \ |
|
227 |
\ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"; |
|
228 |
by (fast_tac FOL_cs 1); |
|
229 |
result(); |
|
230 |
||
231 |
writeln"Problem 27"; |
|
232 |
goal FOL.thy "(EX x. P(x) & ~Q(x)) & \ |
|
233 |
\ (ALL x. P(x) --> R(x)) & \ |
|
234 |
\ (ALL x. M(x) & L(x) --> P(x)) & \ |
|
235 |
\ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \ |
|
236 |
\ --> (ALL x. M(x) --> ~L(x))"; |
|
237 |
by (fast_tac FOL_cs 1); |
|
238 |
result(); |
|
239 |
||
240 |
writeln"Problem 28. AMENDED"; |
|
241 |
goal FOL.thy "(ALL x. P(x) --> (ALL x. Q(x))) & \ |
|
242 |
\ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \ |
|
243 |
\ ((EX x.S(x)) --> (ALL x. L(x) --> M(x))) \ |
|
244 |
\ --> (ALL x. P(x) & L(x) --> M(x))"; |
|
245 |
by (fast_tac FOL_cs 1); |
|
246 |
result(); |
|
247 |
||
248 |
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; |
|
249 |
goal FOL.thy "(EX x. P(x)) & (EX y. Q(y)) \ |
|
250 |
\ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ |
|
251 |
\ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; |
|
252 |
by (fast_tac FOL_cs 1); |
|
253 |
result(); |
|
254 |
||
255 |
writeln"Problem 30"; |
|
256 |
goal FOL.thy "(ALL x. P(x) | Q(x) --> ~ R(x)) & \ |
|
257 |
\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ |
|
258 |
\ --> (ALL x. S(x))"; |
|
259 |
by (fast_tac FOL_cs 1); |
|
260 |
result(); |
|
261 |
||
262 |
writeln"Problem 31"; |
|
263 |
goal FOL.thy "~(EX x.P(x) & (Q(x) | R(x))) & \ |
|
264 |
\ (EX x. L(x) & P(x)) & \ |
|
265 |
\ (ALL x. ~ R(x) --> M(x)) \ |
|
266 |
\ --> (EX x. L(x) & M(x))"; |
|
267 |
by (fast_tac FOL_cs 1); |
|
268 |
result(); |
|
269 |
||
270 |
writeln"Problem 32"; |
|
271 |
goal FOL.thy "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ |
|
272 |
\ (ALL x. S(x) & R(x) --> L(x)) & \ |
|
273 |
\ (ALL x. M(x) --> R(x)) \ |
|
274 |
\ --> (ALL x. P(x) & M(x) --> L(x))"; |
|
275 |
by (best_tac FOL_cs 1); |
|
276 |
result(); |
|
277 |
||
278 |
writeln"Problem 33"; |
|
279 |
goal FOL.thy "(ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \ |
|
280 |
\ (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"; |
|
281 |
by (best_tac FOL_cs 1); |
|
282 |
result(); |
|
283 |
||
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
284 |
writeln"Problem 34 AMENDED (TWICE!!)"; |
0 | 285 |
(*Andrews's challenge*) |
286 |
goal FOL.thy "((EX x. ALL y. p(x) <-> p(y)) <-> \ |
|
287 |
\ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \ |
|
288 |
\ ((EX x. ALL y. q(x) <-> q(y)) <-> \ |
|
289 |
\ ((EX x. p(x)) <-> (ALL y. q(y))))"; |
|
732 | 290 |
by (deepen_tac FOL_cs 0 1); |
0 | 291 |
result(); |
292 |
||
293 |
writeln"Problem 35"; |
|
294 |
goal FOL.thy "EX x y. P(x,y) --> (ALL u v. P(u,v))"; |
|
732 | 295 |
by (mini_tac 1); |
296 |
by (fast_tac FOL_cs 1); |
|
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
297 |
(*Without miniscope, would have to use deepen_tac; would be slower*) |
0 | 298 |
result(); |
299 |
||
300 |
writeln"Problem 36"; |
|
301 |
goal FOL.thy |
|
302 |
"(ALL x. EX y. J(x,y)) & \ |
|
303 |
\ (ALL x. EX y. G(x,y)) & \ |
|
304 |
\ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) \ |
|
305 |
\ --> (ALL x. EX y. H(x,y))"; |
|
306 |
by (fast_tac FOL_cs 1); |
|
307 |
result(); |
|
308 |
||
309 |
writeln"Problem 37"; |
|
310 |
goal FOL.thy "(ALL z. EX w. ALL x. EX y. \ |
|
311 |
\ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u.Q(u,w)))) & \ |
|
312 |
\ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \ |
|
313 |
\ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \ |
|
314 |
\ --> (ALL x. EX y. R(x,y))"; |
|
315 |
by (fast_tac FOL_cs 1); |
|
316 |
(*slow...Poly/ML: 9.7 secs*) |
|
317 |
result(); |
|
318 |
||
428 | 319 |
writeln"Problem 38"; |
0 | 320 |
goal FOL.thy |
321 |
"(ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \ |
|
322 |
\ (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> \ |
|
323 |
\ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \ |
|
324 |
\ (~p(a) | ~(EX y. p(y) & r(x,y)) | \ |
|
325 |
\ (EX z. EX w. p(z) & r(x,w) & r(w,z))))"; |
|
428 | 326 |
by (fast_tac FOL_cs 1); |
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
327 |
result(); |
0 | 328 |
|
329 |
writeln"Problem 39"; |
|
330 |
goal FOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
|
331 |
by (fast_tac FOL_cs 1); |
|
332 |
result(); |
|
333 |
||
334 |
writeln"Problem 40. AMENDED"; |
|
335 |
goal FOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \ |
|
336 |
\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; |
|
337 |
by (fast_tac FOL_cs 1); |
|
338 |
result(); |
|
339 |
||
340 |
writeln"Problem 41"; |
|
341 |
goal FOL.thy "(ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \ |
|
342 |
\ --> ~ (EX z. ALL x. f(x,z))"; |
|
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
343 |
by (fast_tac FOL_cs 1); |
0 | 344 |
result(); |
345 |
||
428 | 346 |
writeln"Problem 42"; |
0 | 347 |
goal FOL.thy "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))"; |
732 | 348 |
by (deepen_tac FOL_cs 0 1); |
428 | 349 |
result(); |
0 | 350 |
|
732 | 351 |
writeln"Problem 43"; |
0 | 352 |
goal FOL.thy "(ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \ |
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
353 |
\ --> (ALL x. ALL y. q(x,y) <-> q(y,x))"; |
732 | 354 |
by (mini_tac 1); |
355 |
by (deepen_tac FOL_cs 5 1); |
|
356 |
(*Faster alternative proof! |
|
357 |
by (asm_simp_tac FOL_ss 1); by (fast_tac FOL_cs 1); |
|
358 |
*) |
|
665
97e9ad6c1c4a
FOL/ex/cla: proofs now use deepen_tac instead of best_tac FOL_dup_cs
lcp
parents:
644
diff
changeset
|
359 |
result(); |
0 | 360 |
|
361 |
writeln"Problem 44"; |
|
362 |
goal FOL.thy "(ALL x. f(x) --> \ |
|
363 |
\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
|
364 |
\ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
|
365 |
\ --> (EX x. j(x) & ~f(x))"; |
|
366 |
by (fast_tac FOL_cs 1); |
|
367 |
result(); |
|
368 |
||
369 |
writeln"Problem 45"; |
|
370 |
goal FOL.thy "(ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \ |
|
371 |
\ --> (ALL y. g(y) & h(x,y) --> k(y))) & \ |
|
372 |
\ ~ (EX y. l(y) & k(y)) & \ |
|
373 |
\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \ |
|
36 | 374 |
\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \ |
0 | 375 |
\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))"; |
376 |
by (best_tac FOL_cs 1); |
|
377 |
result(); |
|
378 |
||
379 |
||
380 |
writeln"Problems (mainly) involving equality or functions"; |
|
381 |
||
382 |
writeln"Problem 48"; |
|
383 |
goal FOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"; |
|
384 |
by (fast_tac FOL_cs 1); |
|
385 |
result(); |
|
386 |
||
387 |
writeln"Problem 49 NOT PROVED AUTOMATICALLY"; |
|
388 |
(*Hard because it involves substitution for Vars; |
|
389 |
the type constraint ensures that x,y,z have the same type as a,b,u. *) |
|
36 | 390 |
goal FOL.thy "(EX x y::'a. ALL z. z=x | z=y) & P(a) & P(b) & a~=b \ |
0 | 391 |
\ --> (ALL u::'a.P(u))"; |
392 |
by (safe_tac FOL_cs); |
|
393 |
by (res_inst_tac [("x","a")] allE 1); |
|
394 |
ba 1; |
|
395 |
by (res_inst_tac [("x","b")] allE 1); |
|
396 |
ba 1; |
|
397 |
by (fast_tac FOL_cs 1); |
|
398 |
result(); |
|
399 |
||
400 |
writeln"Problem 50"; |
|
401 |
(*What has this to do with equality?*) |
|
402 |
goal FOL.thy "(ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))"; |
|
732 | 403 |
by (mini_tac 1); |
404 |
by (deepen_tac FOL_cs 0 1); |
|
0 | 405 |
result(); |
406 |
||
407 |
writeln"Problem 51"; |
|
408 |
goal FOL.thy |
|
409 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
410 |
\ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"; |
|
411 |
by (fast_tac FOL_cs 1); |
|
412 |
result(); |
|
413 |
||
414 |
writeln"Problem 52"; |
|
415 |
(*Almost the same as 51. *) |
|
416 |
goal FOL.thy |
|
417 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
418 |
\ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)"; |
|
419 |
by (best_tac FOL_cs 1); |
|
420 |
result(); |
|
421 |
||
422 |
writeln"Problem 55"; |
|
423 |
||
424 |
(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED |
|
425 |
goal FOL.thy |
|
426 |
"(EX x. lives(x) & killed(x,agatha)) & \ |
|
427 |
\ lives(agatha) & lives(butler) & lives(charles) & \ |
|
428 |
\ (ALL x. lives(x) --> x=agatha | x=butler | x=charles) & \ |
|
429 |
\ (ALL x y. killed(x,y) --> hates(x,y)) & \ |
|
430 |
\ (ALL x y. killed(x,y) --> ~richer(x,y)) & \ |
|
431 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \ |
|
432 |
\ (ALL x. ~ x=butler --> hates(agatha,x)) & \ |
|
433 |
\ (ALL x. ~richer(x,agatha) --> hates(butler,x)) & \ |
|
434 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \ |
|
435 |
\ (ALL x. EX y. ~hates(x,y)) & \ |
|
436 |
\ ~ agatha=butler --> \ |
|
437 |
\ killed(?who,agatha)"; |
|
438 |
by (safe_tac FOL_cs); |
|
439 |
by (dres_inst_tac [("x1","x")] (spec RS mp) 1); |
|
440 |
ba 1; |
|
441 |
be (spec RS exE) 1; |
|
442 |
by (REPEAT (etac allE 1)); |
|
443 |
by (fast_tac FOL_cs 1); |
|
444 |
result(); |
|
445 |
****) |
|
446 |
||
447 |
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988). |
|
448 |
fast_tac DISCOVERS who killed Agatha. *) |
|
449 |
goal FOL.thy "lives(agatha) & lives(butler) & lives(charles) & \ |
|
450 |
\ (killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) & \ |
|
451 |
\ (ALL x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) & \ |
|
452 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \ |
|
453 |
\ (hates(agatha,agatha) & hates(agatha,charles)) & \ |
|
454 |
\ (ALL x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \ |
|
455 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \ |
|
456 |
\ (ALL x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \ |
|
457 |
\ killed(?who,agatha)"; |
|
458 |
by (fast_tac FOL_cs 1); |
|
459 |
result(); |
|
460 |
||
461 |
||
462 |
writeln"Problem 56"; |
|
463 |
goal FOL.thy |
|
464 |
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; |
|
465 |
by (fast_tac FOL_cs 1); |
|
466 |
result(); |
|
467 |
||
468 |
writeln"Problem 57"; |
|
469 |
goal FOL.thy |
|
470 |
"P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
|
471 |
\ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
|
472 |
by (fast_tac FOL_cs 1); |
|
473 |
result(); |
|
474 |
||
475 |
writeln"Problem 58 NOT PROVED AUTOMATICALLY"; |
|
476 |
goal FOL.thy "(ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"; |
|
732 | 477 |
by (slow_tac (FOL_cs addEs [subst_context]) 1); |
0 | 478 |
result(); |
479 |
||
480 |
writeln"Problem 59"; |
|
481 |
goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; |
|
732 | 482 |
by (deepen_tac FOL_cs 0 1); |
0 | 483 |
result(); |
484 |
||
485 |
writeln"Problem 60"; |
|
486 |
goal FOL.thy |
|
487 |
"ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
|
488 |
by (fast_tac FOL_cs 1); |
|
489 |
result(); |
|
490 |
||
1404 | 491 |
writeln"Problem 62 as corrected in AAR newletter #31"; |
492 |
goal FOL.thy |
|
493 |
"(ALL x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <-> \ |
|
494 |
\ (ALL x. (~p(a) | p(x) | p(f(f(x)))) & \ |
|
495 |
\ (~p(a) | ~p(f(x)) | p(f(f(x)))))"; |
|
496 |
by (fast_tac FOL_cs 1); |
|
497 |
result(); |
|
498 |
||
0 | 499 |
|
500 |
writeln"Reached end of file."; |
|
501 |
||
732 | 502 |
(*Thu Jul 23 1992: loaded in 467s using iffE [on SPARC2] *) |
503 |
(*Mon Nov 14 1994: loaded in 144s [on SPARC10, with deepen_tac] *) |
|
504 |
(*Wed Nov 16 1994: loaded in 138s [after addition of norm_term_skip] *) |
|
505 |
(*Mon Nov 21 1994: loaded in 131s [DEPTH_FIRST suppressing repetitions] *) |
|
506 |