author  haftmann 
Thu, 08 Jul 2010 16:19:24 +0200  
changeset 37744  3daaf23b9ab4 
parent 36452  d37c6eed8117 
child 38499  8f0cd11238a7 
permissions  rwrr 
17481  1 
(* Title: LK/LK0.thy 
7093  2 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
3 
Copyright 1993 University of Cambridge 

4 

5 
There may be printing problems if a seqent is in expanded normal form 

35113  6 
(etaexpanded, betacontracted). 
7093  7 
*) 
8 

17481  9 
header {* Classical FirstOrder Sequent Calculus *} 
10 

11 
theory LK0 

12 
imports Sequents 

13 
begin 

7093  14 

15 
global 

16 

17481  17 
classes "term" 
36452  18 
default_sort "term" 
7093  19 

20 
consts 

21 

21524  22 
Trueprop :: "two_seqi" 
7093  23 

17481  24 
True :: o 
25 
False :: o 

22894  26 
equal :: "['a,'a] => o" (infixl "=" 50) 
17481  27 
Not :: "o => o" ("~ _" [40] 40) 
22894  28 
conj :: "[o,o] => o" (infixr "&" 35) 
29 
disj :: "[o,o] => o" (infixr "" 30) 

30 
imp :: "[o,o] => o" (infixr ">" 25) 

31 
iff :: "[o,o] => o" (infixr "<>" 25) 

17481  32 
The :: "('a => o) => 'a" (binder "THE " 10) 
33 
All :: "('a => o) => o" (binder "ALL " 10) 

34 
Ex :: "('a => o) => o" (binder "EX " 10) 

7093  35 

36 
syntax 

35113  37 
"_Trueprop" :: "two_seqe" ("((_)/  (_))" [6,6] 5) 
17481  38 

35113  39 
parse_translation {* [(@{syntax_const "_Trueprop"}, two_seq_tr @{const_syntax Trueprop})] *} 
40 
print_translation {* [(@{const_syntax Trueprop}, two_seq_tr' @{syntax_const "_Trueprop"})] *} 

7093  41 

22894  42 
abbreviation 
43 
not_equal (infixl "~=" 50) where 

44 
"x ~= y == ~ (x = y)" 

7093  45 

35355
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

46 
notation (xsymbols) 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

47 
Not ("\<not> _" [40] 40) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

48 
conj (infixr "\<and>" 35) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

49 
disj (infixr "\<or>" 30) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

50 
imp (infixr "\<longrightarrow>" 25) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

51 
iff (infixr "\<longleftrightarrow>" 25) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

52 
All (binder "\<forall>" 10) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

53 
Ex (binder "\<exists>" 10) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

54 
not_equal (infixl "\<noteq>" 50) 
7093  55 

35355
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

56 
notation (HTML output) 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

57 
Not ("\<not> _" [40] 40) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

58 
conj (infixr "\<and>" 35) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

59 
disj (infixr "\<or>" 30) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

60 
All (binder "\<forall>" 10) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

61 
Ex (binder "\<exists>" 10) and 
613e133966ea
modernized syntax declarations, and make them actually work with authentic syntax;
wenzelm
parents:
35113
diff
changeset

62 
not_equal (infixl "\<noteq>" 50) 
7093  63 

64 
local 

17481  65 

66 
axioms 

7093  67 

68 
(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *) 

69 

17481  70 
contRS: "$H  $E, $S, $S, $F ==> $H  $E, $S, $F" 
71 
contLS: "$H, $S, $S, $G  $E ==> $H, $S, $G  $E" 

7093  72 

17481  73 
thinRS: "$H  $E, $F ==> $H  $E, $S, $F" 
74 
thinLS: "$H, $G  $E ==> $H, $S, $G  $E" 

7093  75 

17481  76 
exchRS: "$H  $E, $R, $S, $F ==> $H  $E, $S, $R, $F" 
77 
exchLS: "$H, $R, $S, $G  $E ==> $H, $S, $R, $G  $E" 

7093  78 

17481  79 
cut: "[ $H  $E, P; $H, P  $E ] ==> $H  $E" 
7093  80 

81 
(*Propositional rules*) 

82 

17481  83 
basic: "$H, P, $G  $E, P, $F" 
7093  84 

17481  85 
conjR: "[ $H $E, P, $F; $H $E, Q, $F ] ==> $H $E, P&Q, $F" 
86 
conjL: "$H, P, Q, $G  $E ==> $H, P & Q, $G  $E" 

7093  87 

17481  88 
disjR: "$H  $E, P, Q, $F ==> $H  $E, PQ, $F" 
89 
disjL: "[ $H, P, $G  $E; $H, Q, $G  $E ] ==> $H, PQ, $G  $E" 

7093  90 

17481  91 
impR: "$H, P  $E, Q, $F ==> $H  $E, P>Q, $F" 
92 
impL: "[ $H,$G  $E,P; $H, Q, $G  $E ] ==> $H, P>Q, $G  $E" 

7093  93 

17481  94 
notR: "$H, P  $E, $F ==> $H  $E, ~P, $F" 
95 
notL: "$H, $G  $E, P ==> $H, ~P, $G  $E" 

7093  96 

17481  97 
FalseL: "$H, False, $G  $E" 
7093  98 

17481  99 
True_def: "True == False>False" 
100 
iff_def: "P<>Q == (P>Q) & (Q>P)" 

7093  101 

102 
(*Quantifiers*) 

103 

17481  104 
allR: "(!!x.$H  $E, P(x), $F) ==> $H  $E, ALL x. P(x), $F" 
105 
allL: "$H, P(x), $G, ALL x. P(x)  $E ==> $H, ALL x. P(x), $G  $E" 

7093  106 

17481  107 
exR: "$H  $E, P(x), $F, EX x. P(x) ==> $H  $E, EX x. P(x), $F" 
108 
exL: "(!!x.$H, P(x), $G  $E) ==> $H, EX x. P(x), $G  $E" 

7093  109 

110 
(*Equality*) 

111 

17481  112 
refl: "$H  $E, a=a, $F" 
113 
subst: "$H(a), $G(a)  $E(a) ==> $H(b), a=b, $G(b)  $E(b)" 

7093  114 

115 
(* Reflection *) 

116 

17481  117 
eq_reflection: " x=y ==> (x==y)" 
118 
iff_reflection: " P<>Q ==> (P==Q)" 

7093  119 

120 
(*Descriptions*) 

121 

17481  122 
The: "[ $H  $E, P(a), $F; !!x.$H, P(x)  $E, x=a, $F ] ==> 
7093  123 
$H  $E, P(THE x. P(x)), $F" 
124 

35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35113
diff
changeset

125 
definition If :: "[o, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) where 
7093  126 
"If(P,x,y) == THE z::'a. (P > z=x) & (~P > z=y)" 
127 

21426  128 

129 
(** Structural Rules on formulas **) 

130 

131 
(*contraction*) 

132 

133 
lemma contR: "$H  $E, P, P, $F ==> $H  $E, P, $F" 

134 
by (rule contRS) 

135 

136 
lemma contL: "$H, P, P, $G  $E ==> $H, P, $G  $E" 

137 
by (rule contLS) 

138 

139 
(*thinning*) 

140 

141 
lemma thinR: "$H  $E, $F ==> $H  $E, P, $F" 

142 
by (rule thinRS) 

143 

144 
lemma thinL: "$H, $G  $E ==> $H, P, $G  $E" 

145 
by (rule thinLS) 

146 

147 
(*exchange*) 

148 

149 
lemma exchR: "$H  $E, Q, P, $F ==> $H  $E, P, Q, $F" 

150 
by (rule exchRS) 

151 

152 
lemma exchL: "$H, Q, P, $G  $E ==> $H, P, Q, $G  $E" 

153 
by (rule exchLS) 

154 

155 
ML {* 

156 
(*Cut and thin, replacing the rightside formula*) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27146
diff
changeset

157 
fun cutR_tac ctxt s i = 
27239  158 
res_inst_tac ctxt [(("P", 0), s) ] @{thm cut} i THEN rtac @{thm thinR} i 
21426  159 

160 
(*Cut and thin, replacing the leftside formula*) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27146
diff
changeset

161 
fun cutL_tac ctxt s i = 
27239  162 
res_inst_tac ctxt [(("P", 0), s)] @{thm cut} i THEN rtac @{thm thinL} (i+1) 
21426  163 
*} 
164 

165 

166 
(** Ifandonlyif rules **) 

167 
lemma iffR: 

168 
"[ $H,P  $E,Q,$F; $H,Q  $E,P,$F ] ==> $H  $E, P <> Q, $F" 

169 
apply (unfold iff_def) 

170 
apply (assumption  rule conjR impR)+ 

171 
done 

172 

173 
lemma iffL: 

174 
"[ $H,$G  $E,P,Q; $H,Q,P,$G  $E ] ==> $H, P <> Q, $G  $E" 

175 
apply (unfold iff_def) 

176 
apply (assumption  rule conjL impL basic)+ 

177 
done 

178 

179 
lemma iff_refl: "$H  $E, (P <> P), $F" 

180 
apply (rule iffR basic)+ 

181 
done 

182 

183 
lemma TrueR: "$H  $E, True, $F" 

184 
apply (unfold True_def) 

185 
apply (rule impR) 

186 
apply (rule basic) 

187 
done 

188 

189 
(*Descriptions*) 

190 
lemma the_equality: 

191 
assumes p1: "$H  $E, P(a), $F" 

192 
and p2: "!!x. $H, P(x)  $E, x=a, $F" 

193 
shows "$H  $E, (THE x. P(x)) = a, $F" 

194 
apply (rule cut) 

195 
apply (rule_tac [2] p2) 

196 
apply (rule The, rule thinR, rule exchRS, rule p1) 

197 
apply (rule thinR, rule exchRS, rule p2) 

198 
done 

199 

200 

201 
(** Weakened quantifier rules. Incomplete, they let the search terminate.**) 

202 

203 
lemma allL_thin: "$H, P(x), $G  $E ==> $H, ALL x. P(x), $G  $E" 

204 
apply (rule allL) 

205 
apply (erule thinL) 

206 
done 

207 

208 
lemma exR_thin: "$H  $E, P(x), $F ==> $H  $E, EX x. P(x), $F" 

209 
apply (rule exR) 

210 
apply (erule thinR) 

211 
done 

212 

213 
(*The rules of LK*) 

214 

215 
ML {* 

216 
val prop_pack = empty_pack add_safes 

217 
[thm "basic", thm "refl", thm "TrueR", thm "FalseL", 

218 
thm "conjL", thm "conjR", thm "disjL", thm "disjR", thm "impL", thm "impR", 

219 
thm "notL", thm "notR", thm "iffL", thm "iffR"]; 

220 

221 
val LK_pack = prop_pack add_safes [thm "allR", thm "exL"] 

222 
add_unsafes [thm "allL_thin", thm "exR_thin", thm "the_equality"]; 

223 

224 
val LK_dup_pack = prop_pack add_safes [thm "allR", thm "exL"] 

225 
add_unsafes [thm "allL", thm "exR", thm "the_equality"]; 

226 

227 

228 
local 

229 
val thinR = thm "thinR" 

230 
val thinL = thm "thinL" 

231 
val cut = thm "cut" 

232 
in 

233 

234 
fun lemma_tac th i = 

235 
rtac (thinR RS cut) i THEN REPEAT (rtac thinL i) THEN rtac th i; 

236 

237 
end; 

238 
*} 

239 

240 
method_setup fast_prop = 

30549  241 
{* Scan.succeed (K (SIMPLE_METHOD' (fast_tac prop_pack))) *} 
21426  242 
"propositional reasoning" 
243 

244 
method_setup fast = 

30549  245 
{* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_pack))) *} 
21426  246 
"classical reasoning" 
247 

248 
method_setup fast_dup = 

30549  249 
{* Scan.succeed (K (SIMPLE_METHOD' (fast_tac LK_dup_pack))) *} 
21426  250 
"classical reasoning" 
251 

252 
method_setup best = 

30549  253 
{* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_pack))) *} 
21426  254 
"classical reasoning" 
255 

256 
method_setup best_dup = 

30549  257 
{* Scan.succeed (K (SIMPLE_METHOD' (best_tac LK_dup_pack))) *} 
21426  258 
"classical reasoning" 
7093  259 

7118
ee384c7b7416
adding missing declarations for the <<...>> notation
paulson
parents:
7093
diff
changeset

260 

21426  261 
lemma mp_R: 
262 
assumes major: "$H  $E, $F, P > Q" 

263 
and minor: "$H  $E, $F, P" 

264 
shows "$H  $E, Q, $F" 

265 
apply (rule thinRS [THEN cut], rule major) 

266 
apply (tactic "step_tac LK_pack 1") 

267 
apply (rule thinR, rule minor) 

268 
done 

269 

270 
lemma mp_L: 

271 
assumes major: "$H, $G  $E, P > Q" 

272 
and minor: "$H, $G, Q  $E" 

273 
shows "$H, P, $G  $E" 

274 
apply (rule thinL [THEN cut], rule major) 

275 
apply (tactic "step_tac LK_pack 1") 

276 
apply (rule thinL, rule minor) 

277 
done 

278 

279 

280 
(** Two rules to generate left and right rules from implications **) 

281 

282 
lemma R_of_imp: 

283 
assumes major: " P > Q" 

284 
and minor: "$H  $E, $F, P" 

285 
shows "$H  $E, Q, $F" 

286 
apply (rule mp_R) 

287 
apply (rule_tac [2] minor) 

288 
apply (rule thinRS, rule major [THEN thinLS]) 

289 
done 

290 

291 
lemma L_of_imp: 

292 
assumes major: " P > Q" 

293 
and minor: "$H, $G, Q  $E" 

294 
shows "$H, P, $G  $E" 

295 
apply (rule mp_L) 

296 
apply (rule_tac [2] minor) 

297 
apply (rule thinRS, rule major [THEN thinLS]) 

298 
done 

299 

300 
(*Can be used to create implications in a subgoal*) 

301 
lemma backwards_impR: 

302 
assumes prem: "$H, $G  $E, $F, P > Q" 

303 
shows "$H, P, $G  $E, Q, $F" 

304 
apply (rule mp_L) 

305 
apply (rule_tac [2] basic) 

306 
apply (rule thinR, rule prem) 

307 
done 

308 

309 
lemma conjunct1: "P&Q ==> P" 

310 
apply (erule thinR [THEN cut]) 

311 
apply fast 

312 
done 

313 

314 
lemma conjunct2: "P&Q ==> Q" 

315 
apply (erule thinR [THEN cut]) 

316 
apply fast 

317 
done 

318 

319 
lemma spec: " (ALL x. P(x)) ==>  P(x)" 

320 
apply (erule thinR [THEN cut]) 

321 
apply fast 

322 
done 

323 

324 

325 
(** Equality **) 

326 

327 
lemma sym: " a=b > b=a" 

328 
by (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *}) 

329 

330 
lemma trans: " a=b > b=c > a=c" 

331 
by (tactic {* safe_tac (LK_pack add_safes [thm "subst"]) 1 *}) 

332 

333 
(* Symmetry of equality in hypotheses *) 

334 
lemmas symL = sym [THEN L_of_imp, standard] 

335 

336 
(* Symmetry of equality in hypotheses *) 

337 
lemmas symR = sym [THEN R_of_imp, standard] 

338 

339 
lemma transR: "[ $H $E, $F, a=b; $H $E, $F, b=c ] ==> $H $E, a=c, $F" 

340 
by (rule trans [THEN R_of_imp, THEN mp_R]) 

341 

342 
(* Two theorms for rewriting only one instance of a definition: 

343 
the first for definitions of formulae and the second for terms *) 

344 

345 
lemma def_imp_iff: "(A == B) ==>  A <> B" 

346 
apply unfold 

347 
apply (rule iff_refl) 

348 
done 

349 

350 
lemma meta_eq_to_obj_eq: "(A == B) ==>  A = B" 

351 
apply unfold 

352 
apply (rule refl) 

353 
done 

354 

355 

356 
(** ifthenelse rules **) 

357 

358 
lemma if_True: " (if True then x else y) = x" 

359 
unfolding If_def by fast 

360 

361 
lemma if_False: " (if False then x else y) = y" 

362 
unfolding If_def by fast 

363 

364 
lemma if_P: " P ==>  (if P then x else y) = x" 

365 
apply (unfold If_def) 

366 
apply (erule thinR [THEN cut]) 

367 
apply fast 

368 
done 

369 

370 
lemma if_not_P: " ~P ==>  (if P then x else y) = y"; 

371 
apply (unfold If_def) 

372 
apply (erule thinR [THEN cut]) 

373 
apply fast 

374 
done 

375 

376 
end 