author  paulson 
Thu, 07 Oct 2004 15:42:30 +0200  
changeset 15234  ec91a90c604e 
parent 15229  1eb23f805c06 
child 15481  fc075ae929e4 
permissions  rwrr 
14770  1 
(* Title: HOL/OrderedGroup.thy 
14738  2 
ID: $Id$ 
14770  3 
Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel 
14738  4 
*) 
5 

6 
header {* Ordered Groups *} 

7 

15131  8 
theory OrderedGroup 
15140  9 
imports Inductive LOrder 
15131  10 
files "../Provers/Arith/abel_cancel.ML" 
11 
begin 

14738  12 

13 
text {* 

14 
The theory of partially ordered groups is taken from the books: 

15 
\begin{itemize} 

16 
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

17 
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 

18 
\end{itemize} 

19 
Most of the used notions can also be looked up in 

20 
\begin{itemize} 

14770  21 
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. 
14738  22 
\item \emph{Algebra I} by van der Waerden, Springer. 
23 
\end{itemize} 

24 
*} 

25 

26 
subsection {* Semigroups, Groups *} 

27 

28 
axclass semigroup_add \<subseteq> plus 

29 
add_assoc: "(a + b) + c = a + (b + c)" 

30 

31 
axclass ab_semigroup_add \<subseteq> semigroup_add 

32 
add_commute: "a + b = b + a" 

33 

34 
lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))" 

35 
by (rule mk_left_commute [of "op +", OF add_assoc add_commute]) 

36 

37 
theorems add_ac = add_assoc add_commute add_left_commute 

38 

39 
axclass semigroup_mult \<subseteq> times 

40 
mult_assoc: "(a * b) * c = a * (b * c)" 

41 

42 
axclass ab_semigroup_mult \<subseteq> semigroup_mult 

43 
mult_commute: "a * b = b * a" 

44 

45 
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))" 

46 
by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute]) 

47 

48 
theorems mult_ac = mult_assoc mult_commute mult_left_commute 

49 

50 
axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add 

51 
add_0[simp]: "0 + a = a" 

52 

53 
axclass monoid_mult \<subseteq> one, semigroup_mult 

54 
mult_1_left[simp]: "1 * a = a" 

55 
mult_1_right[simp]: "a * 1 = a" 

56 

57 
axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult 

58 
mult_1: "1 * a = a" 

59 

60 
instance comm_monoid_mult \<subseteq> monoid_mult 

61 
by (intro_classes, insert mult_1, simp_all add: mult_commute, auto) 

62 

63 
axclass cancel_semigroup_add \<subseteq> semigroup_add 

64 
add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" 

65 
add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" 

66 

67 
axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add 

68 
add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" 

69 

70 
instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add 

71 
proof 

72 
{ 

73 
fix a b c :: 'a 

74 
assume "a + b = a + c" 

75 
thus "b = c" by (rule add_imp_eq) 

76 
} 

77 
note f = this 

78 
fix a b c :: 'a 

79 
assume "b + a = c + a" 

80 
hence "a + b = a + c" by (simp only: add_commute) 

81 
thus "b = c" by (rule f) 

82 
qed 

83 

84 
axclass ab_group_add \<subseteq> minus, comm_monoid_add 

85 
left_minus[simp]: "  a + a = 0" 

86 
diff_minus: "a  b = a + (b)" 

87 

88 
instance ab_group_add \<subseteq> cancel_ab_semigroup_add 

89 
proof 

90 
fix a b c :: 'a 

91 
assume "a + b = a + c" 

92 
hence "a + a + b = a + a + c" by (simp only: add_assoc) 

93 
thus "b = c" by simp 

94 
qed 

95 

96 
lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)" 

97 
proof  

98 
have "a + 0 = 0 + a" by (simp only: add_commute) 

99 
also have "... = a" by simp 

100 
finally show ?thesis . 

101 
qed 

102 

103 
lemma add_left_cancel [simp]: 

104 
"(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))" 

105 
by (blast dest: add_left_imp_eq) 

106 

107 
lemma add_right_cancel [simp]: 

108 
"(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))" 

109 
by (blast dest: add_right_imp_eq) 

110 

111 
lemma right_minus [simp]: "a + (a::'a::ab_group_add) = 0" 

112 
proof  

113 
have "a + a = a + a" by (simp add: add_ac) 

114 
also have "... = 0" by simp 

115 
finally show ?thesis . 

116 
qed 

117 

118 
lemma right_minus_eq: "(a  b = 0) = (a = (b::'a::ab_group_add))" 

119 
proof 

120 
have "a = a  b + b" by (simp add: diff_minus add_ac) 

121 
also assume "a  b = 0" 

122 
finally show "a = b" by simp 

123 
next 

124 
assume "a = b" 

125 
thus "a  b = 0" by (simp add: diff_minus) 

126 
qed 

127 

128 
lemma minus_minus [simp]: " ( (a::'a::ab_group_add)) = a" 

129 
proof (rule add_left_cancel [of "a", THEN iffD1]) 

130 
show "(a + (a) = a + a)" 

131 
by simp 

132 
qed 

133 

134 
lemma equals_zero_I: "a+b = 0 ==> a = (b::'a::ab_group_add)" 

135 
apply (rule right_minus_eq [THEN iffD1, symmetric]) 

136 
apply (simp add: diff_minus add_commute) 

137 
done 

138 

139 
lemma minus_zero [simp]: " 0 = (0::'a::ab_group_add)" 

140 
by (simp add: equals_zero_I) 

141 

142 
lemma diff_self [simp]: "a  (a::'a::ab_group_add) = 0" 

143 
by (simp add: diff_minus) 

144 

145 
lemma diff_0 [simp]: "(0::'a::ab_group_add)  a = a" 

146 
by (simp add: diff_minus) 

147 

148 
lemma diff_0_right [simp]: "a  (0::'a::ab_group_add) = a" 

149 
by (simp add: diff_minus) 

150 

151 
lemma diff_minus_eq_add [simp]: "a   b = a + (b::'a::ab_group_add)" 

152 
by (simp add: diff_minus) 

153 

154 
lemma neg_equal_iff_equal [simp]: "(a = b) = (a = (b::'a::ab_group_add))" 

155 
proof 

156 
assume " a =  b" 

157 
hence " ( a) =  ( b)" 

158 
by simp 

159 
thus "a=b" by simp 

160 
next 

161 
assume "a=b" 

162 
thus "a = b" by simp 

163 
qed 

164 

165 
lemma neg_equal_0_iff_equal [simp]: "(a = 0) = (a = (0::'a::ab_group_add))" 

166 
by (subst neg_equal_iff_equal [symmetric], simp) 

167 

168 
lemma neg_0_equal_iff_equal [simp]: "(0 = a) = (0 = (a::'a::ab_group_add))" 

169 
by (subst neg_equal_iff_equal [symmetric], simp) 

170 

171 
text{*The next two equations can make the simplifier loop!*} 

172 

173 
lemma equation_minus_iff: "(a =  b) = (b =  (a::'a::ab_group_add))" 

174 
proof  

175 
have "( (a) =  b) = ( a = b)" by (rule neg_equal_iff_equal) 

176 
thus ?thesis by (simp add: eq_commute) 

177 
qed 

178 

179 
lemma minus_equation_iff: "( a = b) = ( (b::'a::ab_group_add) = a)" 

180 
proof  

181 
have "( a =  (b)) = (a = b)" by (rule neg_equal_iff_equal) 

182 
thus ?thesis by (simp add: eq_commute) 

183 
qed 

184 

185 
lemma minus_add_distrib [simp]: " (a + b) = a + (b::'a::ab_group_add)" 

186 
apply (rule equals_zero_I) 

187 
apply (simp add: add_ac) 

188 
done 

189 

190 
lemma minus_diff_eq [simp]: " (a  b) = b  (a::'a::ab_group_add)" 

191 
by (simp add: diff_minus add_commute) 

192 

193 
subsection {* (Partially) Ordered Groups *} 

194 

195 
axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add 

196 
add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" 

197 

198 
axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add 

199 

200 
instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add .. 

201 

202 
axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add 

203 
add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" 

204 

205 
axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add 

206 

207 
instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le 

208 
proof 

209 
fix a b c :: 'a 

210 
assume "c + a \<le> c + b" 

211 
hence "(c) + (c + a) \<le> (c) + (c + b)" by (rule add_left_mono) 

212 
hence "((c) + c) + a \<le> ((c) + c) + b" by (simp only: add_assoc) 

213 
thus "a \<le> b" by simp 

214 
qed 

215 

216 
axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder 

217 

218 
instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le 

219 
proof 

220 
fix a b c :: 'a 

221 
assume le: "c + a <= c + b" 

222 
show "a <= b" 

223 
proof (rule ccontr) 

224 
assume w: "~ a \<le> b" 

225 
hence "b <= a" by (simp add: linorder_not_le) 

226 
hence le2: "c+b <= c+a" by (rule add_left_mono) 

227 
have "a = b" 

228 
apply (insert le) 

229 
apply (insert le2) 

230 
apply (drule order_antisym, simp_all) 

231 
done 

232 
with w show False 

233 
by (simp add: linorder_not_le [symmetric]) 

234 
qed 

235 
qed 

236 

237 
lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c" 

238 
by (simp add: add_commute[of _ c] add_left_mono) 

239 

240 
text {* nonstrict, in both arguments *} 

241 
lemma add_mono: 

242 
"[a \<le> b; c \<le> d] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)" 

243 
apply (erule add_right_mono [THEN order_trans]) 

244 
apply (simp add: add_commute add_left_mono) 

245 
done 

246 

247 
lemma add_strict_left_mono: 

248 
"a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)" 

249 
by (simp add: order_less_le add_left_mono) 

250 

251 
lemma add_strict_right_mono: 

252 
"a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)" 

253 
by (simp add: add_commute [of _ c] add_strict_left_mono) 

254 

255 
text{*Strict monotonicity in both arguments*} 

256 
lemma add_strict_mono: "[a<b; c<d] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

257 
apply (erule add_strict_right_mono [THEN order_less_trans]) 

258 
apply (erule add_strict_left_mono) 

259 
done 

260 

261 
lemma add_less_le_mono: 

262 
"[ a<b; c\<le>d ] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

263 
apply (erule add_strict_right_mono [THEN order_less_le_trans]) 

264 
apply (erule add_left_mono) 

265 
done 

266 

267 
lemma add_le_less_mono: 

268 
"[ a\<le>b; c<d ] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

269 
apply (erule add_right_mono [THEN order_le_less_trans]) 

270 
apply (erule add_strict_left_mono) 

271 
done 

272 

273 
lemma add_less_imp_less_left: 

274 
assumes less: "c + a < c + b" shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)" 

275 
proof  

276 
from less have le: "c + a <= c + b" by (simp add: order_le_less) 

277 
have "a <= b" 

278 
apply (insert le) 

279 
apply (drule add_le_imp_le_left) 

280 
by (insert le, drule add_le_imp_le_left, assumption) 

281 
moreover have "a \<noteq> b" 

282 
proof (rule ccontr) 

283 
assume "~(a \<noteq> b)" 

284 
then have "a = b" by simp 

285 
then have "c + a = c + b" by simp 

286 
with less show "False"by simp 

287 
qed 

288 
ultimately show "a < b" by (simp add: order_le_less) 

289 
qed 

290 

291 
lemma add_less_imp_less_right: 

292 
"a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)" 

293 
apply (rule add_less_imp_less_left [of c]) 

294 
apply (simp add: add_commute) 

295 
done 

296 

297 
lemma add_less_cancel_left [simp]: 

298 
"(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" 

299 
by (blast intro: add_less_imp_less_left add_strict_left_mono) 

300 

301 
lemma add_less_cancel_right [simp]: 

302 
"(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))" 

303 
by (blast intro: add_less_imp_less_right add_strict_right_mono) 

304 

305 
lemma add_le_cancel_left [simp]: 

306 
"(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" 

307 
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 

308 

309 
lemma add_le_cancel_right [simp]: 

310 
"(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))" 

311 
by (simp add: add_commute[of a c] add_commute[of b c]) 

312 

313 
lemma add_le_imp_le_right: 

314 
"a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)" 

315 
by simp 

316 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

317 
lemma add_increasing: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

318 
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

319 
shows "[0\<le>a; b\<le>c] ==> b \<le> a + c" 
14738  320 
by (insert add_mono [of 0 a b c], simp) 
321 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

322 
lemma add_strict_increasing: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

323 
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

324 
shows "[0<a; b\<le>c] ==> b < a + c" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

325 
by (insert add_less_le_mono [of 0 a b c], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

326 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

327 
lemma add_strict_increasing2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

328 
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

329 
shows "[0\<le>a; b<c] ==> b < a + c" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

330 
by (insert add_le_less_mono [of 0 a b c], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

331 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

332 

14738  333 
subsection {* Ordering Rules for Unary Minus *} 
334 

335 
lemma le_imp_neg_le: 

336 
assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "b \<le> a" 

337 
proof  

338 
have "a+a \<le> a+b" 

339 
by (rule add_left_mono) 

340 
hence "0 \<le> a+b" 

341 
by simp 

342 
hence "0 + (b) \<le> (a + b) + (b)" 

343 
by (rule add_right_mono) 

344 
thus ?thesis 

345 
by (simp add: add_assoc) 

346 
qed 

347 

348 
lemma neg_le_iff_le [simp]: "(b \<le> a) = (a \<le> (b::'a::pordered_ab_group_add))" 

349 
proof 

350 
assume " b \<le>  a" 

351 
hence " ( a) \<le>  ( b)" 

352 
by (rule le_imp_neg_le) 

353 
thus "a\<le>b" by simp 

354 
next 

355 
assume "a\<le>b" 

356 
thus "b \<le> a" by (rule le_imp_neg_le) 

357 
qed 

358 

359 
lemma neg_le_0_iff_le [simp]: "(a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))" 

360 
by (subst neg_le_iff_le [symmetric], simp) 

361 

362 
lemma neg_0_le_iff_le [simp]: "(0 \<le> a) = (a \<le> (0::'a::pordered_ab_group_add))" 

363 
by (subst neg_le_iff_le [symmetric], simp) 

364 

365 
lemma neg_less_iff_less [simp]: "(b < a) = (a < (b::'a::pordered_ab_group_add))" 

366 
by (force simp add: order_less_le) 

367 

368 
lemma neg_less_0_iff_less [simp]: "(a < 0) = (0 < (a::'a::pordered_ab_group_add))" 

369 
by (subst neg_less_iff_less [symmetric], simp) 

370 

371 
lemma neg_0_less_iff_less [simp]: "(0 < a) = (a < (0::'a::pordered_ab_group_add))" 

372 
by (subst neg_less_iff_less [symmetric], simp) 

373 

374 
text{*The next several equations can make the simplifier loop!*} 

375 

376 
lemma less_minus_iff: "(a <  b) = (b <  (a::'a::pordered_ab_group_add))" 

377 
proof  

378 
have "( (a) <  b) = (b <  a)" by (rule neg_less_iff_less) 

379 
thus ?thesis by simp 

380 
qed 

381 

382 
lemma minus_less_iff: "( a < b) = ( b < (a::'a::pordered_ab_group_add))" 

383 
proof  

384 
have "( a <  (b)) = ( b < a)" by (rule neg_less_iff_less) 

385 
thus ?thesis by simp 

386 
qed 

387 

388 
lemma le_minus_iff: "(a \<le>  b) = (b \<le>  (a::'a::pordered_ab_group_add))" 

389 
proof  

390 
have mm: "!! a (b::'a). ((a)) < b \<Longrightarrow> (b) < a" by (simp only: minus_less_iff) 

391 
have "( ( a) <= b) = (b <=  a)" 

392 
apply (auto simp only: order_le_less) 

393 
apply (drule mm) 

394 
apply (simp_all) 

395 
apply (drule mm[simplified], assumption) 

396 
done 

397 
then show ?thesis by simp 

398 
qed 

399 

400 
lemma minus_le_iff: "( a \<le> b) = ( b \<le> (a::'a::pordered_ab_group_add))" 

401 
by (auto simp add: order_le_less minus_less_iff) 

402 

403 
lemma add_diff_eq: "a + (b  c) = (a + b)  (c::'a::ab_group_add)" 

404 
by (simp add: diff_minus add_ac) 

405 

406 
lemma diff_add_eq: "(a  b) + c = (a + c)  (b::'a::ab_group_add)" 

407 
by (simp add: diff_minus add_ac) 

408 

409 
lemma diff_eq_eq: "(ab = c) = (a = c + (b::'a::ab_group_add))" 

410 
by (auto simp add: diff_minus add_assoc) 

411 

412 
lemma eq_diff_eq: "(a = cb) = (a + (b::'a::ab_group_add) = c)" 

413 
by (auto simp add: diff_minus add_assoc) 

414 

415 
lemma diff_diff_eq: "(a  b)  c = a  (b + (c::'a::ab_group_add))" 

416 
by (simp add: diff_minus add_ac) 

417 

418 
lemma diff_diff_eq2: "a  (b  c) = (a + c)  (b::'a::ab_group_add)" 

419 
by (simp add: diff_minus add_ac) 

420 

421 
lemma diff_add_cancel: "a  b + b = (a::'a::ab_group_add)" 

422 
by (simp add: diff_minus add_ac) 

423 

424 
lemma add_diff_cancel: "a + b  b = (a::'a::ab_group_add)" 

425 
by (simp add: diff_minus add_ac) 

426 

14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

427 
text{*Further subtraction laws*} 
14738  428 

429 
lemma less_iff_diff_less_0: "(a < b) = (a  b < (0::'a::pordered_ab_group_add))" 

430 
proof  

431 
have "(a < b) = (a + ( b) < b + (b))" 

432 
by (simp only: add_less_cancel_right) 

433 
also have "... = (a  b < 0)" by (simp add: diff_minus) 

434 
finally show ?thesis . 

435 
qed 

436 

437 
lemma diff_less_eq: "(ab < c) = (a < c + (b::'a::pordered_ab_group_add))" 

438 
apply (subst less_iff_diff_less_0) 

439 
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) 

440 
apply (simp add: diff_minus add_ac) 

441 
done 

442 

443 
lemma less_diff_eq: "(a < cb) = (a + (b::'a::pordered_ab_group_add) < c)" 

444 
apply (subst less_iff_diff_less_0) 

445 
apply (rule less_iff_diff_less_0 [of _ "cb", THEN ssubst]) 

446 
apply (simp add: diff_minus add_ac) 

447 
done 

448 

449 
lemma diff_le_eq: "(ab \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))" 

450 
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel) 

451 

452 
lemma le_diff_eq: "(a \<le> cb) = (a + (b::'a::pordered_ab_group_add) \<le> c)" 

453 
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel) 

454 

455 
text{*This list of rewrites simplifies (in)equalities by bringing subtractions 

456 
to the top and then moving negative terms to the other side. 

457 
Use with @{text add_ac}*} 

458 
lemmas compare_rls = 

459 
diff_minus [symmetric] 

460 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

461 
diff_less_eq less_diff_eq diff_le_eq le_diff_eq 

462 
diff_eq_eq eq_diff_eq 

463 

464 

465 
subsection{*Lemmas for the @{text cancel_numerals} simproc*} 

466 

467 
lemma eq_iff_diff_eq_0: "(a = b) = (ab = (0::'a::ab_group_add))" 

468 
by (simp add: compare_rls) 

469 

470 
lemma le_iff_diff_le_0: "(a \<le> b) = (ab \<le> (0::'a::pordered_ab_group_add))" 

471 
by (simp add: compare_rls) 

472 

473 
subsection {* Lattice Ordered (Abelian) Groups *} 

474 

475 
axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder 

476 

477 
axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder 

478 

479 
lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))" 

480 
apply (rule order_antisym) 

481 
apply (rule meet_imp_le, simp_all add: meet_join_le) 

482 
apply (rule add_le_imp_le_left [of "a"]) 

483 
apply (simp only: add_assoc[symmetric], simp) 

484 
apply (rule meet_imp_le) 

485 
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ 

486 
done 

487 

488 
lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" 

489 
apply (rule order_antisym) 

490 
apply (rule add_le_imp_le_left [of "a"]) 

491 
apply (simp only: add_assoc[symmetric], simp) 

492 
apply (rule join_imp_le) 

493 
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+ 

494 
apply (rule join_imp_le) 

495 
apply (simp_all add: meet_join_le) 

496 
done 

497 

498 
lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b.  (meet (a) (b)))" 

499 
apply (auto simp add: is_join_def) 

500 
apply (rule_tac c="meet (a) (b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) 

501 
apply (rule_tac c="meet (a) (b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le) 

502 
apply (subst neg_le_iff_le[symmetric]) 

503 
apply (simp add: meet_imp_le) 

504 
done 

505 

506 
lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b.  (join (a) (b)))" 

507 
apply (auto simp add: is_meet_def) 

508 
apply (rule_tac c="join (a) (b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) 

509 
apply (rule_tac c="join (a) (b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le) 

510 
apply (subst neg_le_iff_le[symmetric]) 

511 
apply (simp add: join_imp_le) 

512 
done 

513 

514 
axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder 

515 

516 
instance lordered_ab_group_meet \<subseteq> lordered_ab_group 

517 
proof 

518 
show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet) 

519 
qed 

520 

521 
instance lordered_ab_group_join \<subseteq> lordered_ab_group 

522 
proof 

523 
show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join) 

524 
qed 

525 

526 
lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)" 

527 
proof  

528 
have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left) 

529 
thus ?thesis by (simp add: add_commute) 

530 
qed 

531 

532 
lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)" 

533 
proof  

534 
have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left) 

535 
thus ?thesis by (simp add: add_commute) 

536 
qed 

537 

538 
lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left 

539 

540 
lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) =  meet (a) (b)" 

541 
by (simp add: is_join_unique[OF is_join_join is_join_neg_meet]) 

542 

543 
lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) =  join (a) (b)" 

544 
by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join]) 

545 

546 
lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))" 

547 
proof  

548 
have "0 =  meet 0 (ab) + meet (ab) 0" by (simp add: meet_comm) 

549 
hence "0 = join 0 (ba) + meet (ab) 0" by (simp add: meet_eq_neg_join) 

550 
hence "0 = (a + join a b) + (meet a b + (b))" 

551 
apply (simp add: add_join_distrib_left add_meet_distrib_right) 

552 
by (simp add: diff_minus add_commute) 

553 
thus ?thesis 

554 
apply (simp add: compare_rls) 

555 
apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "a"]) 

556 
apply (simp only: add_assoc, simp add: add_assoc[symmetric]) 

557 
done 

558 
qed 

559 

560 
subsection {* Positive Part, Negative Part, Absolute Value *} 

561 

562 
constdefs 

563 
pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" 

564 
"pprt x == join x 0" 

565 
nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" 

566 
"nprt x == meet x 0" 

567 

568 
lemma prts: "a = pprt a + nprt a" 

569 
by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric]) 

570 

571 
lemma zero_le_pprt[simp]: "0 \<le> pprt a" 

572 
by (simp add: pprt_def meet_join_le) 

573 

574 
lemma nprt_le_zero[simp]: "nprt a \<le> 0" 

575 
by (simp add: nprt_def meet_join_le) 

576 

577 
lemma le_eq_neg: "(a \<le> b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r") 

578 
proof  

579 
have a: "?l \<longrightarrow> ?r" 

580 
apply (auto) 

581 
apply (rule add_le_imp_le_right[of _ "b" _]) 

582 
apply (simp add: add_assoc) 

583 
done 

584 
have b: "?r \<longrightarrow> ?l" 

585 
apply (auto) 

586 
apply (rule add_le_imp_le_right[of _ "b" _]) 

587 
apply (simp) 

588 
done 

589 
from a b show ?thesis by blast 

590 
qed 

591 

592 
lemma join_0_imp_0: "join a (a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" 

593 
proof  

594 
{ 

595 
fix a::'a 

596 
assume hyp: "join a (a) = 0" 

597 
hence "join a (a) + a = a" by (simp) 

598 
hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) 

599 
hence "join (a+a) 0 <= a" by (simp) 

600 
hence "0 <= a" by (blast intro: order_trans meet_join_le) 

601 
} 

602 
note p = this 

603 
assume hyp:"join a (a) = 0" 

604 
hence hyp2:"join (a) ((a)) = 0" by (simp add: join_comm) 

605 
from p[OF hyp] p[OF hyp2] show "a = 0" by simp 

606 
qed 

607 

608 
lemma meet_0_imp_0: "meet a (a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" 

609 
apply (simp add: meet_eq_neg_join) 

610 
apply (simp add: join_comm) 

611 
apply (subst join_0_imp_0) 

612 
by auto 

613 

614 
lemma join_0_eq_0[simp]: "(join a (a) = 0) = (a = (0::'a::lordered_ab_group))" 

615 
by (auto, erule join_0_imp_0) 

616 

617 
lemma meet_0_eq_0[simp]: "(meet a (a) = 0) = (a = (0::'a::lordered_ab_group))" 

618 
by (auto, erule meet_0_imp_0) 

619 

620 
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))" 

621 
proof 

622 
assume "0 <= a + a" 

623 
hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm) 

624 
have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci) 

625 
hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm) 

626 
hence "meet a 0 = 0" by (simp only: add_right_cancel) 

627 
then show "0 <= a" by (simp add: le_def_meet meet_comm) 

628 
next 

629 
assume a: "0 <= a" 

630 
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) 

631 
qed 

632 

633 
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 

634 
proof  

635 
have "(a + a <= 0) = (0 <= (a+a))" by (subst le_minus_iff, simp) 

636 
moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add) 

637 
ultimately show ?thesis by blast 

638 
qed 

639 

640 
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s) 

641 
proof cases 

642 
assume a: "a < 0" 

643 
thus ?s by (simp add: add_strict_mono[OF a a, simplified]) 

644 
next 

645 
assume "~(a < 0)" 

646 
hence a:"0 <= a" by (simp) 

647 
hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified]) 

648 
hence "~(a+a < 0)" by simp 

649 
with a show ?thesis by simp 

650 
qed 

651 

652 
axclass lordered_ab_group_abs \<subseteq> lordered_ab_group 

653 
abs_lattice: "abs x = join x (x)" 

654 

655 
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)" 

656 
by (simp add: abs_lattice) 

657 

658 
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))" 

659 
by (simp add: abs_lattice) 

660 

661 
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))" 

662 
proof  

663 
have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac) 

664 
thus ?thesis by simp 

665 
qed 

666 

667 
lemma neg_meet_eq_join[simp]: " meet a (b::_::lordered_ab_group) = join (a) (b)" 

668 
by (simp add: meet_eq_neg_join) 

669 

670 
lemma neg_join_eq_meet[simp]: " join a (b::_::lordered_ab_group) = meet (a) (b)" 

671 
by (simp del: neg_meet_eq_join add: join_eq_neg_meet) 

672 

673 
lemma join_eq_if: "join a (a) = (if a < 0 then a else (a::'a::{lordered_ab_group, linorder}))" 

674 
proof  

675 
note b = add_le_cancel_right[of a a "a",symmetric,simplified] 

676 
have c: "a + a = 0 \<Longrightarrow> a = a" by (rule add_right_imp_eq[of _ a], simp) 

15197  677 
show ?thesis by (auto simp add: join_max max_def b linorder_not_less) 
14738  678 
qed 
679 

680 
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then a else (a::'a::{lordered_ab_group_abs, linorder}))" 

681 
proof  

682 
show ?thesis by (simp add: abs_lattice join_eq_if) 

683 
qed 

684 

15229  685 
lemma abs_eq [simp]: 
686 
fixes a :: "'a::{lordered_ab_group_abs, linorder}" 

687 
shows "0 \<le> a ==> abs a = a" 

688 
by (simp add: abs_if_lattice linorder_not_less [symmetric]) 

689 

690 
lemma abs_minus_eq [simp]: 

691 
fixes a :: "'a::{lordered_ab_group_abs, linorder}" 

692 
shows "a < 0 ==> abs a = a" 

693 
by (simp add: abs_if_lattice linorder_not_less [symmetric]) 

694 

14738  695 
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)" 
696 
proof  

697 
have a:"a <= abs a" and b:"a <= abs a" by (auto simp add: abs_lattice meet_join_le) 

698 
show ?thesis by (rule add_mono[OF a b, simplified]) 

699 
qed 

700 

701 
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 

702 
proof 

703 
assume "abs a <= 0" 

704 
hence "abs a = 0" by (auto dest: order_antisym) 

705 
thus "a = 0" by simp 

706 
next 

707 
assume "a = 0" 

708 
thus "abs a <= 0" by simp 

709 
qed 

710 

711 
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))" 

712 
by (simp add: order_less_le) 

713 

714 
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)" 

715 
proof  

716 
have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto 

717 
show ?thesis by (simp add: a) 

718 
qed 

719 

720 
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" 

721 
by (simp add: abs_lattice meet_join_le) 

722 

723 
lemma abs_ge_minus_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" 

724 
by (simp add: abs_lattice meet_join_le) 

725 

726 
lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b" 

727 
by (simp add: le_def_join) 

728 

729 
lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a" 

730 
by (simp add: le_def_join join_aci) 

731 

732 
lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a" 

733 
by (simp add: le_def_meet) 

734 

735 
lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b" 

736 
by (simp add: le_def_meet meet_aci) 

737 

738 
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a  nprt a" 

739 
apply (simp add: pprt_def nprt_def diff_minus) 

740 
apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric]) 

741 
apply (subst le_imp_join_eq, auto) 

742 
done 

743 

744 
lemma abs_minus_cancel [simp]: "abs (a) = abs(a::'a::lordered_ab_group_abs)" 

745 
by (simp add: abs_lattice join_comm) 

746 

747 
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)" 

748 
apply (simp add: abs_lattice[of "abs a"]) 

749 
apply (subst ge_imp_join_eq) 

750 
apply (rule order_trans[of _ 0]) 

751 
by auto 

752 

15093
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

753 
lemma abs_minus_commute: 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

754 
fixes a :: "'a::lordered_ab_group_abs" 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

755 
shows "abs (ab) = abs(ba)" 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

756 
proof  
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

757 
have "abs (ab) = abs ( (ab))" by (simp only: abs_minus_cancel) 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

758 
also have "... = abs(ba)" by simp 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

759 
finally show ?thesis . 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

760 
qed 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

761 

14738  762 
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)" 
763 
by (simp add: le_def_meet nprt_def meet_comm) 

764 

765 
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)" 

766 
by (simp add: le_def_join pprt_def join_comm) 

767 

768 
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)" 

769 
by (simp add: le_def_join pprt_def join_comm) 

770 

771 
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)" 

772 
by (simp add: le_def_meet nprt_def meet_comm) 

773 

774 
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)" 

775 
by (simp) 

776 

777 
lemma imp_abs_id: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" 

778 
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts) 

779 

780 
lemma imp_abs_neg_id: "a \<le> 0 \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" 

781 
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts) 

782 

783 
lemma abs_leI: "[a \<le> b; a \<le> b] ==> abs a \<le> (b::'a::lordered_ab_group_abs)" 

784 
by (simp add: abs_lattice join_imp_le) 

785 

786 
lemma le_minus_self_iff: "(a \<le> a) = (a \<le> (0::'a::lordered_ab_group))" 

787 
proof  

788 
from add_le_cancel_left[of "a" "a+a" "0"] have "(a <= a) = (a+a <= 0)" 

789 
by (simp add: add_assoc[symmetric]) 

790 
thus ?thesis by simp 

791 
qed 

792 

793 
lemma minus_le_self_iff: "(a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))" 

794 
proof  

795 
from add_le_cancel_left[of "a" "0" "a+a"] have "(a <= a) = (0 <= a+a)" 

796 
by (simp add: add_assoc[symmetric]) 

797 
thus ?thesis by simp 

798 
qed 

799 

800 
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" 

801 
by (insert abs_ge_self, blast intro: order_trans) 

802 

803 
lemma abs_le_D2: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" 

804 
by (insert abs_le_D1 [of "a"], simp) 

805 

806 
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & a \<le> (b::'a::lordered_ab_group_abs))" 

807 
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) 

808 

809 
lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::lordered_ab_group_abs)" 

810 
proof  

811 
have g:"abs a + abs b = join (a+b) (join (ab) (join (a+b) (a + (b))))" (is "_=join ?m ?n") 

812 
apply (simp add: abs_lattice add_meet_join_distribs join_aci) 

813 
by (simp only: diff_minus) 

814 
have a:"a+b <= join ?m ?n" by (simp add: meet_join_le) 

815 
have b:"ab <= ?n" by (simp add: meet_join_le) 

816 
have c:"?n <= join ?m ?n" by (simp add: meet_join_le) 

817 
from b c have d: "ab <= join ?m ?n" by simp 

818 
have e:"ab = (a+b)" by (simp add: diff_minus) 

819 
from a d e have "abs(a+b) <= join ?m ?n" 

820 
by (drule_tac abs_leI, auto) 

821 
with g[symmetric] show ?thesis by simp 

822 
qed 

823 

824 
lemma abs_diff_triangle_ineq: 

825 
"\<bar>(a::'a::lordered_ab_group_abs) + b  (c+d)\<bar> \<le> \<bar>ac\<bar> + \<bar>bd\<bar>" 

826 
proof  

827 
have "\<bar>a + b  (c+d)\<bar> = \<bar>(ac) + (bd)\<bar>" by (simp add: diff_minus add_ac) 

828 
also have "... \<le> \<bar>ac\<bar> + \<bar>bd\<bar>" by (rule abs_triangle_ineq) 

829 
finally show ?thesis . 

830 
qed 

831 

14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

832 
text {* Needed for abelian cancellation simprocs: *} 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

833 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

834 
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

835 
apply (subst add_left_commute) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

836 
apply (subst add_left_cancel) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

837 
apply simp 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

838 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

839 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

840 
lemma add_cancel_end: "(x + (y + z) = y) = (x =  (z::'a::ab_group_add))" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

841 
apply (subst add_cancel_21[of _ _ _ 0, simplified]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

842 
apply (simp add: add_right_cancel[symmetric, of "x" "z" "z", simplified]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

843 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

844 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

845 
lemma less_eqI: "(x::'a::pordered_ab_group_add)  y = x'  y' \<Longrightarrow> (x < y) = (x' < y')" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

846 
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y']) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

847 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

848 
lemma le_eqI: "(x::'a::pordered_ab_group_add)  y = x'  y' \<Longrightarrow> (y <= x) = (y' <= x')" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

849 
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of y' x']) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

850 
apply (simp add: neg_le_iff_le[symmetric, of "yx" 0] neg_le_iff_le[symmetric, of "y'x'" 0]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

851 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

852 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

853 
lemma eq_eqI: "(x::'a::ab_group_add)  y = x'  y' \<Longrightarrow> (x = y) = (x' = y')" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

854 
by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y']) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

855 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

856 
lemma diff_def: "(x::'a::ab_group_add)  y == x + (y)" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

857 
by (simp add: diff_minus) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

858 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

859 
lemma add_minus_cancel: "(a::'a::ab_group_add) + (a + b) = b" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

860 
by (simp add: add_assoc[symmetric]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

861 

a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

862 
lemma minus_add_cancel: "(a::'a::ab_group_add) + (a + b) = b" 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

863 
by (simp add: add_assoc[symmetric]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

864 

15178  865 
lemma le_add_right_mono: 
866 
assumes 

867 
"a <= b + (c::'a::pordered_ab_group_add)" 

868 
"c <= d" 

869 
shows "a <= b + d" 

870 
apply (rule_tac order_trans[where y = "b+c"]) 

871 
apply (simp_all add: prems) 

872 
done 

873 

874 
lemmas group_eq_simps = 

875 
mult_ac 

876 
add_ac 

877 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

878 
diff_eq_eq eq_diff_eq 

879 

880 
lemma estimate_by_abs: 

881 
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 

882 
proof  

883 
assume 1: "a+b <= c" 

884 
have 2: "a <= c+(b)" 

885 
apply (insert 1) 

886 
apply (drule_tac add_right_mono[where c="b"]) 

887 
apply (simp add: group_eq_simps) 

888 
done 

889 
have 3: "(b) <= abs b" by (rule abs_ge_minus_self) 

890 
show ?thesis by (rule le_add_right_mono[OF 2 3]) 

891 
qed 

892 

893 
lemma abs_of_ge_0: "0 <= (y::'a::lordered_ab_group_abs) \<Longrightarrow> abs y = y" 

894 
proof  

895 
assume 1:"0 <= y" 

896 
have 2:"y <= 0" by (simp add: 1) 

897 
from 1 2 have 3:"y <= y" by (simp only:) 

898 
show ?thesis by (simp add: abs_lattice ge_imp_join_eq[OF 3]) 

899 
qed 

900 

14738  901 
ML {* 
902 
val add_zero_left = thm"add_0"; 

903 
val add_zero_right = thm"add_0_right"; 

904 
*} 

905 

906 
ML {* 

907 
val add_assoc = thm "add_assoc"; 

908 
val add_commute = thm "add_commute"; 

909 
val add_left_commute = thm "add_left_commute"; 

910 
val add_ac = thms "add_ac"; 

911 
val mult_assoc = thm "mult_assoc"; 

912 
val mult_commute = thm "mult_commute"; 

913 
val mult_left_commute = thm "mult_left_commute"; 

914 
val mult_ac = thms "mult_ac"; 

915 
val add_0 = thm "add_0"; 

916 
val mult_1_left = thm "mult_1_left"; 

917 
val mult_1_right = thm "mult_1_right"; 

918 
val mult_1 = thm "mult_1"; 

919 
val add_left_imp_eq = thm "add_left_imp_eq"; 

920 
val add_right_imp_eq = thm "add_right_imp_eq"; 

921 
val add_imp_eq = thm "add_imp_eq"; 

922 
val left_minus = thm "left_minus"; 

923 
val diff_minus = thm "diff_minus"; 

924 
val add_0_right = thm "add_0_right"; 

925 
val add_left_cancel = thm "add_left_cancel"; 

926 
val add_right_cancel = thm "add_right_cancel"; 

927 
val right_minus = thm "right_minus"; 

928 
val right_minus_eq = thm "right_minus_eq"; 

929 
val minus_minus = thm "minus_minus"; 

930 
val equals_zero_I = thm "equals_zero_I"; 

931 
val minus_zero = thm "minus_zero"; 

932 
val diff_self = thm "diff_self"; 

933 
val diff_0 = thm "diff_0"; 

934 
val diff_0_right = thm "diff_0_right"; 

935 
val diff_minus_eq_add = thm "diff_minus_eq_add"; 

936 
val neg_equal_iff_equal = thm "neg_equal_iff_equal"; 

937 
val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal"; 

938 
val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal"; 

939 
val equation_minus_iff = thm "equation_minus_iff"; 

940 
val minus_equation_iff = thm "minus_equation_iff"; 

941 
val minus_add_distrib = thm "minus_add_distrib"; 

942 
val minus_diff_eq = thm "minus_diff_eq"; 

943 
val add_left_mono = thm "add_left_mono"; 

944 
val add_le_imp_le_left = thm "add_le_imp_le_left"; 

945 
val add_right_mono = thm "add_right_mono"; 

946 
val add_mono = thm "add_mono"; 

947 
val add_strict_left_mono = thm "add_strict_left_mono"; 

948 
val add_strict_right_mono = thm "add_strict_right_mono"; 

949 
val add_strict_mono = thm "add_strict_mono"; 

950 
val add_less_le_mono = thm "add_less_le_mono"; 

951 
val add_le_less_mono = thm "add_le_less_mono"; 

952 
val add_less_imp_less_left = thm "add_less_imp_less_left"; 

953 
val add_less_imp_less_right = thm "add_less_imp_less_right"; 

954 
val add_less_cancel_left = thm "add_less_cancel_left"; 

955 
val add_less_cancel_right = thm "add_less_cancel_right"; 

956 
val add_le_cancel_left = thm "add_le_cancel_left"; 

957 
val add_le_cancel_right = thm "add_le_cancel_right"; 

958 
val add_le_imp_le_right = thm "add_le_imp_le_right"; 

959 
val add_increasing = thm "add_increasing"; 

960 
val le_imp_neg_le = thm "le_imp_neg_le"; 

961 
val neg_le_iff_le = thm "neg_le_iff_le"; 

962 
val neg_le_0_iff_le = thm "neg_le_0_iff_le"; 

963 
val neg_0_le_iff_le = thm "neg_0_le_iff_le"; 

964 
val neg_less_iff_less = thm "neg_less_iff_less"; 

965 
val neg_less_0_iff_less = thm "neg_less_0_iff_less"; 

966 
val neg_0_less_iff_less = thm "neg_0_less_iff_less"; 

967 
val less_minus_iff = thm "less_minus_iff"; 

968 
val minus_less_iff = thm "minus_less_iff"; 

969 
val le_minus_iff = thm "le_minus_iff"; 

970 
val minus_le_iff = thm "minus_le_iff"; 

971 
val add_diff_eq = thm "add_diff_eq"; 

972 
val diff_add_eq = thm "diff_add_eq"; 

973 
val diff_eq_eq = thm "diff_eq_eq"; 

974 
val eq_diff_eq = thm "eq_diff_eq"; 

975 
val diff_diff_eq = thm "diff_diff_eq"; 

976 
val diff_diff_eq2 = thm "diff_diff_eq2"; 

977 
val diff_add_cancel = thm "diff_add_cancel"; 

978 
val add_diff_cancel = thm "add_diff_cancel"; 

979 
val less_iff_diff_less_0 = thm "less_iff_diff_less_0"; 

980 
val diff_less_eq = thm "diff_less_eq"; 

981 
val less_diff_eq = thm "less_diff_eq"; 

982 
val diff_le_eq = thm "diff_le_eq"; 

983 
val le_diff_eq = thm "le_diff_eq"; 

984 
val compare_rls = thms "compare_rls"; 

985 
val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0"; 

986 
val le_iff_diff_le_0 = thm "le_iff_diff_le_0"; 

987 
val add_meet_distrib_left = thm "add_meet_distrib_left"; 

988 
val add_join_distrib_left = thm "add_join_distrib_left"; 

989 
val is_join_neg_meet = thm "is_join_neg_meet"; 

990 
val is_meet_neg_join = thm "is_meet_neg_join"; 

991 
val add_join_distrib_right = thm "add_join_distrib_right"; 

992 
val add_meet_distrib_right = thm "add_meet_distrib_right"; 

993 
val add_meet_join_distribs = thms "add_meet_join_distribs"; 

994 
val join_eq_neg_meet = thm "join_eq_neg_meet"; 

995 
val meet_eq_neg_join = thm "meet_eq_neg_join"; 

996 
val add_eq_meet_join = thm "add_eq_meet_join"; 

997 
val prts = thm "prts"; 

998 
val zero_le_pprt = thm "zero_le_pprt"; 

999 
val nprt_le_zero = thm "nprt_le_zero"; 

1000 
val le_eq_neg = thm "le_eq_neg"; 

1001 
val join_0_imp_0 = thm "join_0_imp_0"; 

1002 
val meet_0_imp_0 = thm "meet_0_imp_0"; 

1003 
val join_0_eq_0 = thm "join_0_eq_0"; 

1004 
val meet_0_eq_0 = thm "meet_0_eq_0"; 

1005 
val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add"; 

1006 
val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero"; 

1007 
val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero"; 

1008 
val abs_lattice = thm "abs_lattice"; 

1009 
val abs_zero = thm "abs_zero"; 

1010 
val abs_eq_0 = thm "abs_eq_0"; 

1011 
val abs_0_eq = thm "abs_0_eq"; 

1012 
val neg_meet_eq_join = thm "neg_meet_eq_join"; 

1013 
val neg_join_eq_meet = thm "neg_join_eq_meet"; 

1014 
val join_eq_if = thm "join_eq_if"; 

1015 
val abs_if_lattice = thm "abs_if_lattice"; 

1016 
val abs_ge_zero = thm "abs_ge_zero"; 

1017 
val abs_le_zero_iff = thm "abs_le_zero_iff"; 

1018 
val zero_less_abs_iff = thm "zero_less_abs_iff"; 

1019 
val abs_not_less_zero = thm "abs_not_less_zero"; 

1020 
val abs_ge_self = thm "abs_ge_self"; 

1021 
val abs_ge_minus_self = thm "abs_ge_minus_self"; 

1022 
val le_imp_join_eq = thm "le_imp_join_eq"; 

1023 
val ge_imp_join_eq = thm "ge_imp_join_eq"; 

1024 
val le_imp_meet_eq = thm "le_imp_meet_eq"; 

1025 
val ge_imp_meet_eq = thm "ge_imp_meet_eq"; 

1026 
val abs_prts = thm "abs_prts"; 

1027 
val abs_minus_cancel = thm "abs_minus_cancel"; 

1028 
val abs_idempotent = thm "abs_idempotent"; 

1029 
val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt"; 

1030 
val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt"; 

1031 
val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id"; 

1032 
val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id"; 

1033 
val iff2imp = thm "iff2imp"; 

1034 
val imp_abs_id = thm "imp_abs_id"; 

1035 
val imp_abs_neg_id = thm "imp_abs_neg_id"; 

1036 
val abs_leI = thm "abs_leI"; 

1037 
val le_minus_self_iff = thm "le_minus_self_iff"; 

1038 
val minus_le_self_iff = thm "minus_le_self_iff"; 

1039 
val abs_le_D1 = thm "abs_le_D1"; 

1040 
val abs_le_D2 = thm "abs_le_D2"; 

1041 
val abs_le_iff = thm "abs_le_iff"; 

1042 
val abs_triangle_ineq = thm "abs_triangle_ineq"; 

1043 
val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq"; 

1044 
*} 

1045 

1046 
end 