author  nipkow 
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changeset 21312  1d39091a3208 
parent 21245  23e6eb4d0975 
child 21328  73bb86d0f483 
permissions  rwrr 
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(* Title: HOL/OrderedGroup.thy 
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ID: $Id$ 
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel, 
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with contributions by Jeremy Avigad 
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*) 
6 

7 
header {* Ordered Groups *} 

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15131  9 
theory OrderedGroup 
15140  10 
imports Inductive LOrder 
19798  11 
uses "~~/src/Provers/Arith/abel_cancel.ML" 
15131  12 
begin 
14738  13 

14 
text {* 

15 
The theory of partially ordered groups is taken from the books: 

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\begin{itemize} 

17 
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

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

19 
\end{itemize} 

20 
Most of the used notions can also be looked up in 

21 
\begin{itemize} 

14770  22 
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. 
14738  23 
\item \emph{Algebra I} by van der Waerden, Springer. 
24 
\end{itemize} 

25 
*} 

26 

27 
subsection {* Semigroups, Groups *} 

28 

29 
axclass semigroup_add \<subseteq> plus 

30 
add_assoc: "(a + b) + c = a + (b + c)" 

31 

32 
axclass ab_semigroup_add \<subseteq> semigroup_add 

33 
add_commute: "a + b = b + a" 

34 

35 
lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))" 

36 
by (rule mk_left_commute [of "op +", OF add_assoc add_commute]) 

37 

38 
theorems add_ac = add_assoc add_commute add_left_commute 

39 

40 
axclass semigroup_mult \<subseteq> times 

41 
mult_assoc: "(a * b) * c = a * (b * c)" 

42 

43 
axclass ab_semigroup_mult \<subseteq> semigroup_mult 

44 
mult_commute: "a * b = b * a" 

45 

46 
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))" 

47 
by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute]) 

48 

49 
theorems mult_ac = mult_assoc mult_commute mult_left_commute 

50 

51 
axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add 

52 
add_0[simp]: "0 + a = a" 

53 

54 
axclass monoid_mult \<subseteq> one, semigroup_mult 

55 
mult_1_left[simp]: "1 * a = a" 

56 
mult_1_right[simp]: "a * 1 = a" 

57 

58 
axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult 

59 
mult_1: "1 * a = a" 

60 

61 
instance comm_monoid_mult \<subseteq> monoid_mult 

62 
by (intro_classes, insert mult_1, simp_all add: mult_commute, auto) 

63 

64 
axclass cancel_semigroup_add \<subseteq> semigroup_add 

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

66 
add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" 

67 

68 
axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add 

69 
add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" 

70 

71 
instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add 

72 
proof 

73 
{ 

74 
fix a b c :: 'a 

75 
assume "a + b = a + c" 

76 
thus "b = c" by (rule add_imp_eq) 

77 
} 

78 
note f = this 

79 
fix a b c :: 'a 

80 
assume "b + a = c + a" 

81 
hence "a + b = a + c" by (simp only: add_commute) 

82 
thus "b = c" by (rule f) 

83 
qed 

84 

85 
axclass ab_group_add \<subseteq> minus, comm_monoid_add 

86 
left_minus[simp]: "  a + a = 0" 

87 
diff_minus: "a  b = a + (b)" 

88 

89 
instance ab_group_add \<subseteq> cancel_ab_semigroup_add 

90 
proof 

91 
fix a b c :: 'a 

92 
assume "a + b = a + c" 

93 
hence "a + a + b = a + a + c" by (simp only: add_assoc) 

94 
thus "b = c" by simp 

95 
qed 

96 

97 
lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)" 

98 
proof  

99 
have "a + 0 = 0 + a" by (simp only: add_commute) 

100 
also have "... = a" by simp 

101 
finally show ?thesis . 

102 
qed 

103 

21245  104 
lemmas add_zero_left = add_0 
105 
and add_zero_right = add_0_right 

106 

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lemma add_left_cancel [simp]: 
108 
"(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))" 

109 
by (blast dest: add_left_imp_eq) 

110 

111 
lemma add_right_cancel [simp]: 

112 
"(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))" 

113 
by (blast dest: add_right_imp_eq) 

114 

115 
lemma right_minus [simp]: "a + (a::'a::ab_group_add) = 0" 

116 
proof  

117 
have "a + a = a + a" by (simp add: add_ac) 

118 
also have "... = 0" by simp 

119 
finally show ?thesis . 

120 
qed 

121 

122 
lemma right_minus_eq: "(a  b = 0) = (a = (b::'a::ab_group_add))" 

123 
proof 

124 
have "a = a  b + b" by (simp add: diff_minus add_ac) 

125 
also assume "a  b = 0" 

126 
finally show "a = b" by simp 

127 
next 

128 
assume "a = b" 

129 
thus "a  b = 0" by (simp add: diff_minus) 

130 
qed 

131 

132 
lemma minus_minus [simp]: " ( (a::'a::ab_group_add)) = a" 

133 
proof (rule add_left_cancel [of "a", THEN iffD1]) 

134 
show "(a + (a) = a + a)" 

135 
by simp 

136 
qed 

137 

138 
lemma equals_zero_I: "a+b = 0 ==> a = (b::'a::ab_group_add)" 

139 
apply (rule right_minus_eq [THEN iffD1, symmetric]) 

140 
apply (simp add: diff_minus add_commute) 

141 
done 

142 

143 
lemma minus_zero [simp]: " 0 = (0::'a::ab_group_add)" 

144 
by (simp add: equals_zero_I) 

145 

146 
lemma diff_self [simp]: "a  (a::'a::ab_group_add) = 0" 

147 
by (simp add: diff_minus) 

148 

149 
lemma diff_0 [simp]: "(0::'a::ab_group_add)  a = a" 

150 
by (simp add: diff_minus) 

151 

152 
lemma diff_0_right [simp]: "a  (0::'a::ab_group_add) = a" 

153 
by (simp add: diff_minus) 

154 

155 
lemma diff_minus_eq_add [simp]: "a   b = a + (b::'a::ab_group_add)" 

156 
by (simp add: diff_minus) 

157 

158 
lemma neg_equal_iff_equal [simp]: "(a = b) = (a = (b::'a::ab_group_add))" 

159 
proof 

160 
assume " a =  b" 

161 
hence " ( a) =  ( b)" 

162 
by simp 

163 
thus "a=b" by simp 

164 
next 

165 
assume "a=b" 

166 
thus "a = b" by simp 

167 
qed 

168 

169 
lemma neg_equal_0_iff_equal [simp]: "(a = 0) = (a = (0::'a::ab_group_add))" 

170 
by (subst neg_equal_iff_equal [symmetric], simp) 

171 

172 
lemma neg_0_equal_iff_equal [simp]: "(0 = a) = (0 = (a::'a::ab_group_add))" 

173 
by (subst neg_equal_iff_equal [symmetric], simp) 

174 

175 
text{*The next two equations can make the simplifier loop!*} 

176 

177 
lemma equation_minus_iff: "(a =  b) = (b =  (a::'a::ab_group_add))" 

178 
proof  

179 
have "( (a) =  b) = ( a = b)" by (rule neg_equal_iff_equal) 

180 
thus ?thesis by (simp add: eq_commute) 

181 
qed 

182 

183 
lemma minus_equation_iff: "( a = b) = ( (b::'a::ab_group_add) = a)" 

184 
proof  

185 
have "( a =  (b)) = (a = b)" by (rule neg_equal_iff_equal) 

186 
thus ?thesis by (simp add: eq_commute) 

187 
qed 

188 

189 
lemma minus_add_distrib [simp]: " (a + b) = a + (b::'a::ab_group_add)" 

190 
apply (rule equals_zero_I) 

191 
apply (simp add: add_ac) 

192 
done 

193 

194 
lemma minus_diff_eq [simp]: " (a  b) = b  (a::'a::ab_group_add)" 

195 
by (simp add: diff_minus add_commute) 

196 

197 
subsection {* (Partially) Ordered Groups *} 

198 

199 
axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add 

200 
add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" 

201 

202 
axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add 

203 

204 
instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add .. 

205 

206 
axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add 

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

208 

209 
axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add 

210 

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instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le 

212 
proof 

213 
fix a b c :: 'a 

214 
assume "c + a \<le> c + b" 

215 
hence "(c) + (c + a) \<le> (c) + (c + b)" by (rule add_left_mono) 

216 
hence "((c) + c) + a \<le> ((c) + c) + b" by (simp only: add_assoc) 

217 
thus "a \<le> b" by simp 

218 
qed 

219 

220 
axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder 

221 

222 
instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le 

223 
proof 

224 
fix a b c :: 'a 

225 
assume le: "c + a <= c + b" 

226 
show "a <= b" 

227 
proof (rule ccontr) 

228 
assume w: "~ a \<le> b" 

229 
hence "b <= a" by (simp add: linorder_not_le) 

230 
hence le2: "c+b <= c+a" by (rule add_left_mono) 

231 
have "a = b" 

232 
apply (insert le) 

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apply (insert le2) 

234 
apply (drule order_antisym, simp_all) 

235 
done 

236 
with w show False 

237 
by (simp add: linorder_not_le [symmetric]) 

238 
qed 

239 
qed 

240 

241 
lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c" 

242 
by (simp add: add_commute[of _ c] add_left_mono) 

243 

244 
text {* nonstrict, in both arguments *} 

245 
lemma add_mono: 

246 
"[a \<le> b; c \<le> d] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)" 

247 
apply (erule add_right_mono [THEN order_trans]) 

248 
apply (simp add: add_commute add_left_mono) 

249 
done 

250 

251 
lemma add_strict_left_mono: 

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

253 
by (simp add: order_less_le add_left_mono) 

254 

255 
lemma add_strict_right_mono: 

256 
"a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)" 

257 
by (simp add: add_commute [of _ c] add_strict_left_mono) 

258 

259 
text{*Strict monotonicity in both arguments*} 

260 
lemma add_strict_mono: "[a<b; c<d] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

261 
apply (erule add_strict_right_mono [THEN order_less_trans]) 

262 
apply (erule add_strict_left_mono) 

263 
done 

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265 
lemma add_less_le_mono: 

266 
"[ a<b; c\<le>d ] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

267 
apply (erule add_strict_right_mono [THEN order_less_le_trans]) 

268 
apply (erule add_left_mono) 

269 
done 

270 

271 
lemma add_le_less_mono: 

272 
"[ a\<le>b; c<d ] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)" 

273 
apply (erule add_right_mono [THEN order_le_less_trans]) 

274 
apply (erule add_strict_left_mono) 

275 
done 

276 

277 
lemma add_less_imp_less_left: 

278 
assumes less: "c + a < c + b" shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)" 

279 
proof  

280 
from less have le: "c + a <= c + b" by (simp add: order_le_less) 

281 
have "a <= b" 

282 
apply (insert le) 

283 
apply (drule add_le_imp_le_left) 

284 
by (insert le, drule add_le_imp_le_left, assumption) 

285 
moreover have "a \<noteq> b" 

286 
proof (rule ccontr) 

287 
assume "~(a \<noteq> b)" 

288 
then have "a = b" by simp 

289 
then have "c + a = c + b" by simp 

290 
with less show "False"by simp 

291 
qed 

292 
ultimately show "a < b" by (simp add: order_le_less) 

293 
qed 

294 

295 
lemma add_less_imp_less_right: 

296 
"a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)" 

297 
apply (rule add_less_imp_less_left [of c]) 

298 
apply (simp add: add_commute) 

299 
done 

300 

301 
lemma add_less_cancel_left [simp]: 

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

303 
by (blast intro: add_less_imp_less_left add_strict_left_mono) 

304 

305 
lemma add_less_cancel_right [simp]: 

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

307 
by (blast intro: add_less_imp_less_right add_strict_right_mono) 

308 

309 
lemma add_le_cancel_left [simp]: 

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

311 
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 

312 

313 
lemma add_le_cancel_right [simp]: 

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

315 
by (simp add: add_commute[of a c] add_commute[of b c]) 

316 

317 
lemma add_le_imp_le_right: 

318 
"a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)" 

319 
by simp 

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lemma add_increasing: 
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fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
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shows "[0\<le>a; b\<le>c] ==> b \<le> a + c" 
14738  324 
by (insert add_mono [of 0 a b c], simp) 
325 

15539  326 
lemma add_increasing2: 
327 
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 

328 
shows "[0\<le>c; b\<le>a] ==> b \<le> a + c" 

329 
by (simp add:add_increasing add_commute[of a]) 

330 

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lemma add_strict_increasing: 
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fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
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shows "[0<a; b\<le>c] ==> b < a + c" 
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by (insert add_less_le_mono [of 0 a b c], simp) 
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lemma add_strict_increasing2: 
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fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" 
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shows "[0\<le>a; b<c] ==> b < a + c" 
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by (insert add_le_less_mono [of 0 a b c], simp) 
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19527  341 
lemma max_add_distrib_left: 
342 
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" 

343 
shows "(max x y) + z = max (x+z) (y+z)" 

344 
by (rule max_of_mono [THEN sym], rule add_le_cancel_right) 

345 

346 
lemma min_add_distrib_left: 

347 
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" 

348 
shows "(min x y) + z = min (x+z) (y+z)" 

349 
by (rule min_of_mono [THEN sym], rule add_le_cancel_right) 

350 

351 
lemma max_diff_distrib_left: 

352 
fixes z :: "'a::pordered_ab_group_add" 

353 
shows "(max x y)  z = max (xz) (yz)" 

354 
by (simp add: diff_minus, rule max_add_distrib_left) 

355 

356 
lemma min_diff_distrib_left: 

357 
fixes z :: "'a::pordered_ab_group_add" 

358 
shows "(min x y)  z = min (xz) (yz)" 

359 
by (simp add: diff_minus, rule min_add_distrib_left) 

360 

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14738  362 
subsection {* Ordering Rules for Unary Minus *} 
363 

364 
lemma le_imp_neg_le: 

365 
assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "b \<le> a" 

366 
proof  

367 
have "a+a \<le> a+b" 

368 
by (rule add_left_mono) 

369 
hence "0 \<le> a+b" 

370 
by simp 

371 
hence "0 + (b) \<le> (a + b) + (b)" 

372 
by (rule add_right_mono) 

373 
thus ?thesis 

374 
by (simp add: add_assoc) 

375 
qed 

376 

377 
lemma neg_le_iff_le [simp]: "(b \<le> a) = (a \<le> (b::'a::pordered_ab_group_add))" 

378 
proof 

379 
assume " b \<le>  a" 

380 
hence " ( a) \<le>  ( b)" 

381 
by (rule le_imp_neg_le) 

382 
thus "a\<le>b" by simp 

383 
next 

384 
assume "a\<le>b" 

385 
thus "b \<le> a" by (rule le_imp_neg_le) 

386 
qed 

387 

388 
lemma neg_le_0_iff_le [simp]: "(a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))" 

389 
by (subst neg_le_iff_le [symmetric], simp) 

390 

391 
lemma neg_0_le_iff_le [simp]: "(0 \<le> a) = (a \<le> (0::'a::pordered_ab_group_add))" 

392 
by (subst neg_le_iff_le [symmetric], simp) 

393 

394 
lemma neg_less_iff_less [simp]: "(b < a) = (a < (b::'a::pordered_ab_group_add))" 

395 
by (force simp add: order_less_le) 

396 

397 
lemma neg_less_0_iff_less [simp]: "(a < 0) = (0 < (a::'a::pordered_ab_group_add))" 

398 
by (subst neg_less_iff_less [symmetric], simp) 

399 

400 
lemma neg_0_less_iff_less [simp]: "(0 < a) = (a < (0::'a::pordered_ab_group_add))" 

401 
by (subst neg_less_iff_less [symmetric], simp) 

402 

403 
text{*The next several equations can make the simplifier loop!*} 

404 

405 
lemma less_minus_iff: "(a <  b) = (b <  (a::'a::pordered_ab_group_add))" 

406 
proof  

407 
have "( (a) <  b) = (b <  a)" by (rule neg_less_iff_less) 

408 
thus ?thesis by simp 

409 
qed 

410 

411 
lemma minus_less_iff: "( a < b) = ( b < (a::'a::pordered_ab_group_add))" 

412 
proof  

413 
have "( a <  (b)) = ( b < a)" by (rule neg_less_iff_less) 

414 
thus ?thesis by simp 

415 
qed 

416 

417 
lemma le_minus_iff: "(a \<le>  b) = (b \<le>  (a::'a::pordered_ab_group_add))" 

418 
proof  

419 
have mm: "!! a (b::'a). ((a)) < b \<Longrightarrow> (b) < a" by (simp only: minus_less_iff) 

420 
have "( ( a) <= b) = (b <=  a)" 

421 
apply (auto simp only: order_le_less) 

422 
apply (drule mm) 

423 
apply (simp_all) 

424 
apply (drule mm[simplified], assumption) 

425 
done 

426 
then show ?thesis by simp 

427 
qed 

428 

429 
lemma minus_le_iff: "( a \<le> b) = ( b \<le> (a::'a::pordered_ab_group_add))" 

430 
by (auto simp add: order_le_less minus_less_iff) 

431 

432 
lemma add_diff_eq: "a + (b  c) = (a + b)  (c::'a::ab_group_add)" 

433 
by (simp add: diff_minus add_ac) 

434 

435 
lemma diff_add_eq: "(a  b) + c = (a + c)  (b::'a::ab_group_add)" 

436 
by (simp add: diff_minus add_ac) 

437 

438 
lemma diff_eq_eq: "(ab = c) = (a = c + (b::'a::ab_group_add))" 

439 
by (auto simp add: diff_minus add_assoc) 

440 

441 
lemma eq_diff_eq: "(a = cb) = (a + (b::'a::ab_group_add) = c)" 

442 
by (auto simp add: diff_minus add_assoc) 

443 

444 
lemma diff_diff_eq: "(a  b)  c = a  (b + (c::'a::ab_group_add))" 

445 
by (simp add: diff_minus add_ac) 

446 

447 
lemma diff_diff_eq2: "a  (b  c) = (a + c)  (b::'a::ab_group_add)" 

448 
by (simp add: diff_minus add_ac) 

449 

450 
lemma diff_add_cancel: "a  b + b = (a::'a::ab_group_add)" 

451 
by (simp add: diff_minus add_ac) 

452 

453 
lemma add_diff_cancel: "a + b  b = (a::'a::ab_group_add)" 

454 
by (simp add: diff_minus add_ac) 

455 

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

456 
text{*Further subtraction laws*} 
14738  457 

458 
lemma less_iff_diff_less_0: "(a < b) = (a  b < (0::'a::pordered_ab_group_add))" 

459 
proof  

460 
have "(a < b) = (a + ( b) < b + (b))" 

461 
by (simp only: add_less_cancel_right) 

462 
also have "... = (a  b < 0)" by (simp add: diff_minus) 

463 
finally show ?thesis . 

464 
qed 

465 

466 
lemma diff_less_eq: "(ab < c) = (a < c + (b::'a::pordered_ab_group_add))" 

15481  467 
apply (subst less_iff_diff_less_0 [of a]) 
14738  468 
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) 
469 
apply (simp add: diff_minus add_ac) 

470 
done 

471 

472 
lemma less_diff_eq: "(a < cb) = (a + (b::'a::pordered_ab_group_add) < c)" 

15481  473 
apply (subst less_iff_diff_less_0 [of "a+b"]) 
474 
apply (subst less_iff_diff_less_0 [of a]) 

14738  475 
apply (simp add: diff_minus add_ac) 
476 
done 

477 

478 
lemma diff_le_eq: "(ab \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))" 

479 
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel) 

480 

481 
lemma le_diff_eq: "(a \<le> cb) = (a + (b::'a::pordered_ab_group_add) \<le> c)" 

482 
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel) 

483 

484 
text{*This list of rewrites simplifies (in)equalities by bringing subtractions 

485 
to the top and then moving negative terms to the other side. 

486 
Use with @{text add_ac}*} 

487 
lemmas compare_rls = 

488 
diff_minus [symmetric] 

489 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

490 
diff_less_eq less_diff_eq diff_le_eq le_diff_eq 

491 
diff_eq_eq eq_diff_eq 

492 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

493 
subsection {* Support for reasoning about signs *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

494 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

495 
lemma add_pos_pos: "0 < 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

496 
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

497 
==> 0 < y ==> 0 < x + y" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

498 
apply (subgoal_tac "0 + 0 < x + y") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

499 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

500 
apply (erule add_less_le_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

501 
apply (erule order_less_imp_le) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

502 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

503 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

504 
lemma add_pos_nonneg: "0 < 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

505 
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

506 
==> 0 <= y ==> 0 < x + y" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

507 
apply (subgoal_tac "0 + 0 < x + y") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

508 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

509 
apply (erule add_less_le_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

510 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

511 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

512 
lemma add_nonneg_pos: "0 <= 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

513 
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

514 
==> 0 < y ==> 0 < x + y" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

515 
apply (subgoal_tac "0 + 0 < x + y") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

516 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

517 
apply (erule add_le_less_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

518 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

519 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

520 
lemma add_nonneg_nonneg: "0 <= 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

521 
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

522 
==> 0 <= y ==> 0 <= x + y" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

523 
apply (subgoal_tac "0 + 0 <= x + y") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

524 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

525 
apply (erule add_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

526 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

527 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

528 
lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

529 
< 0 ==> y < 0 ==> x + y < 0" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

530 
apply (subgoal_tac "x + y < 0 + 0") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

531 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

532 
apply (erule add_less_le_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

533 
apply (erule order_less_imp_le) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

534 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

535 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

536 
lemma add_neg_nonpos: 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

537 
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

538 
==> y <= 0 ==> x + y < 0" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

539 
apply (subgoal_tac "x + y < 0 + 0") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

540 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

541 
apply (erule add_less_le_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

542 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

543 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

544 
lemma add_nonpos_neg: 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

545 
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

546 
==> y < 0 ==> x + y < 0" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

547 
apply (subgoal_tac "x + y < 0 + 0") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

548 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

549 
apply (erule add_le_less_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

550 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

551 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

552 
lemma add_nonpos_nonpos: 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

553 
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

554 
==> y <= 0 ==> x + y <= 0" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

555 
apply (subgoal_tac "x + y <= 0 + 0") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

556 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

557 
apply (erule add_mono, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

558 
done 
14738  559 

560 
subsection{*Lemmas for the @{text cancel_numerals} simproc*} 

561 

562 
lemma eq_iff_diff_eq_0: "(a = b) = (ab = (0::'a::ab_group_add))" 

563 
by (simp add: compare_rls) 

564 

565 
lemma le_iff_diff_le_0: "(a \<le> b) = (ab \<le> (0::'a::pordered_ab_group_add))" 

566 
by (simp add: compare_rls) 

567 

568 
subsection {* Lattice Ordered (Abelian) Groups *} 

569 

570 
axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder 

571 

572 
axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder 

573 

574 
lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))" 

575 
apply (rule order_antisym) 

21312  576 
apply (simp_all add: le_meetI) 
14738  577 
apply (rule add_le_imp_le_left [of "a"]) 
578 
apply (simp only: add_assoc[symmetric], simp) 

21312  579 
apply rule 
580 
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+ 

14738  581 
done 
582 

583 
lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))" 

584 
apply (rule order_antisym) 

585 
apply (rule add_le_imp_le_left [of "a"]) 

586 
apply (simp only: add_assoc[symmetric], simp) 

21312  587 
apply rule 
588 
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+ 

589 
apply (rule join_leI) 

590 
apply (simp_all) 

14738  591 
done 
592 

593 
lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b.  (meet (a) (b)))" 

594 
apply (auto simp add: is_join_def) 

21312  595 
apply (rule_tac c="meet (a) (b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left) 
596 
apply (rule_tac c="meet (a) (b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left) 

14738  597 
apply (subst neg_le_iff_le[symmetric]) 
21312  598 
apply (simp add: le_meetI) 
14738  599 
done 
600 

601 
lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b.  (join (a) (b)))" 

602 
apply (auto simp add: is_meet_def) 

21312  603 
apply (rule_tac c="join (a) (b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left) 
604 
apply (rule_tac c="join (a) (b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left) 

14738  605 
apply (subst neg_le_iff_le[symmetric]) 
21312  606 
apply (simp add: join_leI) 
14738  607 
done 
608 

609 
axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder 

610 

611 
instance lordered_ab_group_meet \<subseteq> lordered_ab_group 

612 
proof 

613 
show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet) 

614 
qed 

615 

616 
instance lordered_ab_group_join \<subseteq> lordered_ab_group 

617 
proof 

618 
show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join) 

619 
qed 

620 

621 
lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)" 

622 
proof  

623 
have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left) 

624 
thus ?thesis by (simp add: add_commute) 

625 
qed 

626 

627 
lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)" 

628 
proof  

629 
have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left) 

630 
thus ?thesis by (simp add: add_commute) 

631 
qed 

632 

633 
lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left 

634 

635 
lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) =  meet (a) (b)" 

636 
by (simp add: is_join_unique[OF is_join_join is_join_neg_meet]) 

637 

638 
lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) =  join (a) (b)" 

639 
by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join]) 

640 

641 
lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))" 

642 
proof  

643 
have "0 =  meet 0 (ab) + meet (ab) 0" by (simp add: meet_comm) 

644 
hence "0 = join 0 (ba) + meet (ab) 0" by (simp add: meet_eq_neg_join) 

645 
hence "0 = (a + join a b) + (meet a b + (b))" 

646 
apply (simp add: add_join_distrib_left add_meet_distrib_right) 

647 
by (simp add: diff_minus add_commute) 

648 
thus ?thesis 

649 
apply (simp add: compare_rls) 

650 
apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "a"]) 

651 
apply (simp only: add_assoc, simp add: add_assoc[symmetric]) 

652 
done 

653 
qed 

654 

655 
subsection {* Positive Part, Negative Part, Absolute Value *} 

656 

657 
constdefs 

658 
pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" 

659 
"pprt x == join x 0" 

660 
nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" 

661 
"nprt x == meet x 0" 

662 

663 
lemma prts: "a = pprt a + nprt a" 

664 
by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric]) 

665 

666 
lemma zero_le_pprt[simp]: "0 \<le> pprt a" 

21312  667 
by (simp add: pprt_def) 
14738  668 

669 
lemma nprt_le_zero[simp]: "nprt a \<le> 0" 

21312  670 
by (simp add: nprt_def) 
14738  671 

672 
lemma le_eq_neg: "(a \<le> b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r") 

673 
proof  

674 
have a: "?l \<longrightarrow> ?r" 

675 
apply (auto) 

676 
apply (rule add_le_imp_le_right[of _ "b" _]) 

677 
apply (simp add: add_assoc) 

678 
done 

679 
have b: "?r \<longrightarrow> ?l" 

680 
apply (auto) 

681 
apply (rule add_le_imp_le_right[of _ "b" _]) 

682 
apply (simp) 

683 
done 

684 
from a b show ?thesis by blast 

685 
qed 

686 

15580  687 
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def) 
688 
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def) 

689 

690 
lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x" 

691 
by (simp add: pprt_def le_def_join join_aci) 

692 

693 
lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x" 

694 
by (simp add: nprt_def le_def_meet meet_aci) 

695 

696 
lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0" 

697 
by (simp add: pprt_def le_def_join join_aci) 

698 

699 
lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0" 

700 
by (simp add: nprt_def le_def_meet meet_aci) 

701 

14738  702 
lemma join_0_imp_0: "join a (a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" 
703 
proof  

704 
{ 

705 
fix a::'a 

706 
assume hyp: "join a (a) = 0" 

707 
hence "join a (a) + a = a" by (simp) 

708 
hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right) 

709 
hence "join (a+a) 0 <= a" by (simp) 

710 
hence "0 <= a" by (blast intro: order_trans meet_join_le) 

711 
} 

712 
note p = this 

713 
assume hyp:"join a (a) = 0" 

714 
hence hyp2:"join (a) ((a)) = 0" by (simp add: join_comm) 

715 
from p[OF hyp] p[OF hyp2] show "a = 0" by simp 

716 
qed 

717 

718 
lemma meet_0_imp_0: "meet a (a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" 

719 
apply (simp add: meet_eq_neg_join) 

720 
apply (simp add: join_comm) 

15481  721 
apply (erule join_0_imp_0) 
722 
done 

14738  723 

724 
lemma join_0_eq_0[simp]: "(join a (a) = 0) = (a = (0::'a::lordered_ab_group))" 

725 
by (auto, erule join_0_imp_0) 

726 

727 
lemma meet_0_eq_0[simp]: "(meet a (a) = 0) = (a = (0::'a::lordered_ab_group))" 

728 
by (auto, erule meet_0_imp_0) 

729 

730 
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))" 

731 
proof 

732 
assume "0 <= a + a" 

733 
hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm) 

734 
have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci) 

735 
hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm) 

736 
hence "meet a 0 = 0" by (simp only: add_right_cancel) 

737 
then show "0 <= a" by (simp add: le_def_meet meet_comm) 

738 
next 

739 
assume a: "0 <= a" 

740 
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) 

741 
qed 

742 

743 
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 

744 
proof  

745 
have "(a + a <= 0) = (0 <= (a+a))" by (subst le_minus_iff, simp) 

746 
moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add) 

747 
ultimately show ?thesis by blast 

748 
qed 

749 

750 
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s) 

751 
proof cases 

752 
assume a: "a < 0" 

753 
thus ?s by (simp add: add_strict_mono[OF a a, simplified]) 

754 
next 

755 
assume "~(a < 0)" 

756 
hence a:"0 <= a" by (simp) 

757 
hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified]) 

758 
hence "~(a+a < 0)" by simp 

759 
with a show ?thesis by simp 

760 
qed 

761 

762 
axclass lordered_ab_group_abs \<subseteq> lordered_ab_group 

763 
abs_lattice: "abs x = join x (x)" 

764 

765 
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)" 

766 
by (simp add: abs_lattice) 

767 

768 
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))" 

769 
by (simp add: abs_lattice) 

770 

771 
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))" 

772 
proof  

773 
have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac) 

774 
thus ?thesis by simp 

775 
qed 

776 

777 
lemma neg_meet_eq_join[simp]: " meet a (b::_::lordered_ab_group) = join (a) (b)" 

778 
by (simp add: meet_eq_neg_join) 

779 

780 
lemma neg_join_eq_meet[simp]: " join a (b::_::lordered_ab_group) = meet (a) (b)" 

781 
by (simp del: neg_meet_eq_join add: join_eq_neg_meet) 

782 

783 
lemma join_eq_if: "join a (a) = (if a < 0 then a else (a::'a::{lordered_ab_group, linorder}))" 

784 
proof  

785 
note b = add_le_cancel_right[of a a "a",symmetric,simplified] 

786 
have c: "a + a = 0 \<Longrightarrow> a = a" by (rule add_right_imp_eq[of _ a], simp) 

15197  787 
show ?thesis by (auto simp add: join_max max_def b linorder_not_less) 
14738  788 
qed 
789 

790 
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then a else (a::'a::{lordered_ab_group_abs, linorder}))" 

791 
proof  

792 
show ?thesis by (simp add: abs_lattice join_eq_if) 

793 
qed 

794 

795 
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)" 

796 
proof  

21312  797 
have a:"a <= abs a" and b:"a <= abs a" by (auto simp add: abs_lattice) 
14738  798 
show ?thesis by (rule add_mono[OF a b, simplified]) 
799 
qed 

800 

801 
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 

802 
proof 

803 
assume "abs a <= 0" 

804 
hence "abs a = 0" by (auto dest: order_antisym) 

805 
thus "a = 0" by simp 

806 
next 

807 
assume "a = 0" 

808 
thus "abs a <= 0" by simp 

809 
qed 

810 

811 
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))" 

812 
by (simp add: order_less_le) 

813 

814 
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)" 

815 
proof  

816 
have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto 

817 
show ?thesis by (simp add: a) 

818 
qed 

819 

820 
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" 

21312  821 
by (simp add: abs_lattice) 
14738  822 

823 
lemma abs_ge_minus_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" 

21312  824 
by (simp add: abs_lattice) 
14738  825 

826 
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a  nprt a" 

827 
apply (simp add: pprt_def nprt_def diff_minus) 

828 
apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric]) 

21312  829 
apply (subst join_absorp2, auto) 
14738  830 
done 
831 

832 
lemma abs_minus_cancel [simp]: "abs (a) = abs(a::'a::lordered_ab_group_abs)" 

833 
by (simp add: abs_lattice join_comm) 

834 

835 
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)" 

836 
apply (simp add: abs_lattice[of "abs a"]) 

21312  837 
apply (subst join_absorp1) 
14738  838 
apply (rule order_trans[of _ 0]) 
839 
by auto 

840 

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

841 
lemma abs_minus_commute: 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

842 
fixes a :: "'a::lordered_ab_group_abs" 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

843 
shows "abs (ab) = abs(ba)" 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

844 
proof  
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

845 
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

846 
also have "... = abs(ba)" by simp 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

847 
finally show ?thesis . 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

848 
qed 
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset

849 

14738  850 
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)" 
851 
by (simp add: le_def_meet nprt_def meet_comm) 

852 

853 
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)" 

854 
by (simp add: le_def_join pprt_def join_comm) 

855 

856 
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)" 

857 
by (simp add: le_def_join pprt_def join_comm) 

858 

859 
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)" 

860 
by (simp add: le_def_meet nprt_def meet_comm) 

861 

15580  862 
lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b" 
863 
by (simp add: le_def_join pprt_def join_aci) 

864 

865 
lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b" 

866 
by (simp add: le_def_meet nprt_def meet_aci) 

867 

19404  868 
lemma pprt_neg: "pprt (x) =  nprt x" 
869 
by (simp add: pprt_def nprt_def) 

870 

871 
lemma nprt_neg: "nprt (x) =  pprt x" 

872 
by (simp add: pprt_def nprt_def) 

873 

14738  874 
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)" 
875 
by (simp) 

876 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

877 
lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" 
14738  878 
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts) 
879 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

880 
lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x"; 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

881 
by (rule abs_of_nonneg, rule order_less_imp_le); 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

882 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

883 
lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" 
14738  884 
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts) 
885 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

886 
lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) < 0 ==> 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

887 
abs x =  x" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

888 
by (rule abs_of_nonpos, rule order_less_imp_le) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

889 

14738  890 
lemma abs_leI: "[a \<le> b; a \<le> b] ==> abs a \<le> (b::'a::lordered_ab_group_abs)" 
21312  891 
by (simp add: abs_lattice join_leI) 
14738  892 

893 
lemma le_minus_self_iff: "(a \<le> a) = (a \<le> (0::'a::lordered_ab_group))" 

894 
proof  

895 
from add_le_cancel_left[of "a" "a+a" "0"] have "(a <= a) = (a+a <= 0)" 

896 
by (simp add: add_assoc[symmetric]) 

897 
thus ?thesis by simp 

898 
qed 

899 

900 
lemma minus_le_self_iff: "(a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))" 

901 
proof  

902 
from add_le_cancel_left[of "a" "0" "a+a"] have "(a <= a) = (0 <= a+a)" 

903 
by (simp add: add_assoc[symmetric]) 

904 
thus ?thesis by simp 

905 
qed 

906 

907 
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" 

908 
by (insert abs_ge_self, blast intro: order_trans) 

909 

910 
lemma abs_le_D2: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" 

911 
by (insert abs_le_D1 [of "a"], simp) 

912 

913 
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & a \<le> (b::'a::lordered_ab_group_abs))" 

914 
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) 

915 

15539  916 
lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)" 
14738  917 
proof  
918 
have g:"abs a + abs b = join (a+b) (join (ab) (join (a+b) (a + (b))))" (is "_=join ?m ?n") 

19233
77ca20b0ed77
renamed HOL +  * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
17085
diff
changeset

919 
by (simp add: abs_lattice add_meet_join_distribs join_aci diff_minus) 
21312  920 
have a:"a+b <= join ?m ?n" by (simp) 
921 
have b:"ab <= ?n" by (simp) 

922 
have c:"?n <= join ?m ?n" by (simp) 

923 
from b c have d: "ab <= join ?m ?n" by(rule order_trans) 

14738  924 
have e:"ab = (a+b)" by (simp add: diff_minus) 
925 
from a d e have "abs(a+b) <= join ?m ?n" 

926 
by (drule_tac abs_leI, auto) 

927 
with g[symmetric] show ?thesis by simp 

928 
qed 

929 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

930 
lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs)  
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

931 
abs b <= abs (a  b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

932 
apply (simp add: compare_rls) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

933 
apply (subgoal_tac "abs a = abs (a  b + b)") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

934 
apply (erule ssubst) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

935 
apply (rule abs_triangle_ineq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

936 
apply (rule arg_cong);back; 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

937 
apply (simp add: compare_rls) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

938 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

939 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

940 
lemma abs_triangle_ineq3: 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

941 
"abs(abs (a::'a::lordered_ab_group_abs)  abs b) <= abs (a  b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

942 
apply (subst abs_le_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

943 
apply auto 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

944 
apply (rule abs_triangle_ineq2) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

945 
apply (subst abs_minus_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

946 
apply (rule abs_triangle_ineq2) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

947 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

948 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

949 
lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs)  b) <= 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

950 
abs a + abs b" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

951 
proof ; 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

952 
have "abs(a  b) = abs(a +  b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

953 
by (subst diff_minus, rule refl) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

954 
also have "... <= abs a + abs ( b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

955 
by (rule abs_triangle_ineq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

956 
finally show ?thesis 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

957 
by simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

958 
qed 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

959 

14738  960 
lemma abs_diff_triangle_ineq: 
961 
"\<bar>(a::'a::lordered_ab_group_abs) + b  (c+d)\<bar> \<le> \<bar>ac\<bar> + \<bar>bd\<bar>" 

962 
proof  

963 
have "\<bar>a + b  (c+d)\<bar> = \<bar>(ac) + (bd)\<bar>" by (simp add: diff_minus add_ac) 

964 
also have "... \<le> \<bar>ac\<bar> + \<bar>bd\<bar>" by (rule abs_triangle_ineq) 

965 
finally show ?thesis . 

966 
qed 

967 

15539  968 
lemma abs_add_abs[simp]: 
969 
fixes a:: "'a::{lordered_ab_group_abs}" 

970 
shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R") 

971 
proof (rule order_antisym) 

972 
show "?L \<ge> ?R" by(rule abs_ge_self) 

973 
next 

974 
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq) 

975 
also have "\<dots> = ?R" by simp 

976 
finally show "?L \<le> ?R" . 

977 
qed 

978 

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

979 
text {* Needed for abelian cancellation simprocs: *} 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

980 

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

981 
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

982 
apply (subst add_left_commute) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

983 
apply (subst add_left_cancel) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

984 
apply simp 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

985 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

986 

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

987 
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

988 
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

989 
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

990 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

991 

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

992 
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

993 
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

994 

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

995 
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

996 
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

997 
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

998 
done 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

999 

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

1000 
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

1001 
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

1002 

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

1003 
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

1004 
by (simp add: diff_minus) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

1005 

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

1006 
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

1007 
by (simp add: add_assoc[symmetric]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

1008 

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

1009 
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

1010 
by (simp add: add_assoc[symmetric]) 
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

1011 

15178  1012 
lemma le_add_right_mono: 
1013 
assumes 

1014 
"a <= b + (c::'a::pordered_ab_group_add)" 

1015 
"c <= d" 

1016 
shows "a <= b + d" 

1017 
apply (rule_tac order_trans[where y = "b+c"]) 

1018 
apply (simp_all add: prems) 

1019 
done 

1020 

1021 
lemmas group_eq_simps = 

1022 
mult_ac 

1023 
add_ac 

1024 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

1025 
diff_eq_eq eq_diff_eq 

1026 

1027 
lemma estimate_by_abs: 

1028 
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 

1029 
proof  

1030 
assume 1: "a+b <= c" 

1031 
have 2: "a <= c+(b)" 

1032 
apply (insert 1) 

1033 
apply (drule_tac add_right_mono[where c="b"]) 

1034 
apply (simp add: group_eq_simps) 

1035 
done 

1036 
have 3: "(b) <= abs b" by (rule abs_ge_minus_self) 

1037 
show ?thesis by (rule le_add_right_mono[OF 2 3]) 

1038 
qed 

1039 

17085  1040 
text{*Simplification of @{term "xy < 0"}, etc.*} 
1041 
lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric] 

1042 
lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric] 

1043 
lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric] 

1044 
declare diff_less_0_iff_less [simp] 

1045 
declare diff_eq_0_iff_eq [simp] 

1046 
declare diff_le_0_iff_le [simp] 

1047 

1048 

14738  1049 
ML {* 
1050 
val add_assoc = thm "add_assoc"; 

1051 
val add_commute = thm "add_commute"; 

1052 
val add_left_commute = thm "add_left_commute"; 

1053 
val add_ac = thms "add_ac"; 

1054 
val mult_assoc = thm "mult_assoc"; 

1055 
val mult_commute = thm "mult_commute"; 

1056 
val mult_left_commute = thm "mult_left_commute"; 

1057 
val mult_ac = thms "mult_ac"; 

1058 
val add_0 = thm "add_0"; 

1059 
val mult_1_left = thm "mult_1_left"; 

1060 
val mult_1_right = thm "mult_1_right"; 

1061 
val mult_1 = thm "mult_1"; 

1062 
val add_left_imp_eq = thm "add_left_imp_eq"; 

1063 
val add_right_imp_eq = thm "add_right_imp_eq"; 

1064 
val add_imp_eq = thm "add_imp_eq"; 

1065 
val left_minus = thm "left_minus"; 

1066 
val diff_minus = thm "diff_minus"; 

1067 
val add_0_right = thm "add_0_right"; 

1068 
val add_left_cancel = thm "add_left_cancel"; 

1069 
val add_right_cancel = thm "add_right_cancel"; 

1070 
val right_minus = thm "right_minus"; 

1071 
val right_minus_eq = thm "right_minus_eq"; 

1072 
val minus_minus = thm "minus_minus"; 

1073 
val equals_zero_I = thm "equals_zero_I"; 

1074 
val minus_zero = thm "minus_zero"; 

1075 
val diff_self = thm "diff_self"; 

1076 
val diff_0 = thm "diff_0"; 

1077 
val diff_0_right = thm "diff_0_right"; 

1078 
val diff_minus_eq_add = thm "diff_minus_eq_add"; 

1079 
val neg_equal_iff_equal = thm "neg_equal_iff_equal"; 

1080 
val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal"; 

1081 
val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal"; 

1082 
val equation_minus_iff = thm "equation_minus_iff"; 

1083 
val minus_equation_iff = thm "minus_equation_iff"; 

1084 
val minus_add_distrib = thm "minus_add_distrib"; 

1085 
val minus_diff_eq = thm "minus_diff_eq"; 

1086 
val add_left_mono = thm "add_left_mono"; 

1087 
val add_le_imp_le_left = thm "add_le_imp_le_left"; 

1088 
val add_right_mono = thm "add_right_mono"; 

1089 
val add_mono = thm "add_mono"; 

1090 
val add_strict_left_mono = thm "add_strict_left_mono"; 

1091 
val add_strict_right_mono = thm "add_strict_right_mono"; 

1092 
val add_strict_mono = thm "add_strict_mono"; 

1093 
val add_less_le_mono = thm "add_less_le_mono"; 

1094 
val add_le_less_mono = thm "add_le_less_mono"; 

1095 
val add_less_imp_less_left = thm "add_less_imp_less_left"; 

1096 
val add_less_imp_less_right = thm "add_less_imp_less_right"; 

1097 
val add_less_cancel_left = thm "add_less_cancel_left"; 

1098 
val add_less_cancel_right = thm "add_less_cancel_right"; 

1099 
val add_le_cancel_left = thm "add_le_cancel_left"; 

1100 
val add_le_cancel_right = thm "add_le_cancel_right"; 

1101 
val add_le_imp_le_right = thm "add_le_imp_le_right"; 

1102 
val add_increasing = thm "add_increasing"; 

1103 
val le_imp_neg_le = thm "le_imp_neg_le"; 

1104 
val neg_le_iff_le = thm "neg_le_iff_le"; 

1105 
val neg_le_0_iff_le = thm "neg_le_0_iff_le"; 

1106 
val neg_0_le_iff_le = thm "neg_0_le_iff_le"; 

1107 
val neg_less_iff_less = thm "neg_less_iff_less"; 

1108 
val neg_less_0_iff_less = thm "neg_less_0_iff_less"; 

1109 
val neg_0_less_iff_less = thm "neg_0_less_iff_less"; 

1110 
val less_minus_iff = thm "less_minus_iff"; 

1111 
val minus_less_iff = thm "minus_less_iff"; 

1112 
val le_minus_iff = thm "le_minus_iff"; 

1113 
val minus_le_iff = thm "minus_le_iff"; 

1114 
val add_diff_eq = thm "add_diff_eq"; 

1115 
val diff_add_eq = thm "diff_add_eq"; 

1116 
val diff_eq_eq = thm "diff_eq_eq"; 

1117 
val eq_diff_eq = thm "eq_diff_eq"; 

1118 
val diff_diff_eq = thm "diff_diff_eq"; 

1119 
val diff_diff_eq2 = thm "diff_diff_eq2"; 

1120 
val diff_add_cancel = thm "diff_add_cancel"; 

1121 
val add_diff_cancel = thm "add_diff_cancel"; 

1122 
val less_iff_diff_less_0 = thm "less_iff_diff_less_0"; 

1123 
val diff_less_eq = thm "diff_less_eq"; 

1124 
val less_diff_eq = thm "less_diff_eq"; 

1125 
val diff_le_eq = thm "diff_le_eq"; 

1126 
val le_diff_eq = thm "le_diff_eq"; 

1127 
val compare_rls = thms "compare_rls"; 

1128 
val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0"; 

1129 
val le_iff_diff_le_0 = thm "le_iff_diff_le_0"; 

1130 
val add_meet_distrib_left = thm "add_meet_distrib_left"; 

1131 
val add_join_distrib_left = thm "add_join_distrib_left"; 

1132 
val is_join_neg_meet = thm "is_join_neg_meet"; 

1133 
val is_meet_neg_join = thm "is_meet_neg_join"; 

1134 
val add_join_distrib_right = thm "add_join_distrib_right"; 

1135 
val add_meet_distrib_right = thm "add_meet_distrib_right"; 

1136 
val add_meet_join_distribs = thms "add_meet_join_distribs"; 

1137 
val join_eq_neg_meet = thm "join_eq_neg_meet"; 

1138 
val meet_eq_neg_join = thm "meet_eq_neg_join"; 

1139 
val add_eq_meet_join = thm "add_eq_meet_join"; 

1140 
val prts = thm "prts"; 

1141 
val zero_le_pprt = thm "zero_le_pprt"; 

1142 
val nprt_le_zero = thm "nprt_le_zero"; 

1143 
val le_eq_neg = thm "le_eq_neg"; 

1144 
val join_0_imp_0 = thm "join_0_imp_0"; 

1145 
val meet_0_imp_0 = thm "meet_0_imp_0"; 

1146 
val join_0_eq_0 = thm "join_0_eq_0"; 

1147 
val meet_0_eq_0 = thm "meet_0_eq_0"; 

1148 
val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add"; 

1149 
val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero"; 

1150 
val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero"; 

1151 
val abs_lattice = thm "abs_lattice"; 

1152 
val abs_zero = thm "abs_zero"; 

1153 
val abs_eq_0 = thm "abs_eq_0"; 

1154 
val abs_0_eq = thm "abs_0_eq"; 

1155 
val neg_meet_eq_join = thm "neg_meet_eq_join"; 

1156 
val neg_join_eq_meet = thm "neg_join_eq_meet"; 

1157 
val join_eq_if = thm "join_eq_if"; 

1158 
val abs_if_lattice = thm "abs_if_lattice"; 

1159 
val abs_ge_zero = thm "abs_ge_zero"; 

1160 
val abs_le_zero_iff = thm "abs_le_zero_iff"; 

1161 
val zero_less_abs_iff = thm "zero_less_abs_iff"; 

1162 
val abs_not_less_zero = thm "abs_not_less_zero"; 

1163 
val abs_ge_self = thm "abs_ge_self"; 

1164 
val abs_ge_minus_self = thm "abs_ge_minus_self"; 

21312  1165 
val le_imp_join_eq = thm "join_absorp2"; 
1166 
val ge_imp_join_eq = thm "join_absorp1"; 

1167 
val le_imp_meet_eq = thm "meet_absorp1"; 

1168 
val ge_imp_meet_eq = thm "meet_absorp2"; 

14738  1169 
val abs_prts = thm "abs_prts"; 
1170 
val abs_minus_cancel = thm "abs_minus_cancel"; 

1171 
val abs_idempotent = thm "abs_idempotent"; 

1172 
val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt"; 

1173 
val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt"; 

1174 
val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id"; 

1175 
val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id"; 

1176 
val iff2imp = thm "iff2imp"; 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

1177 
(* val imp_abs_id = thm "imp_abs_id"; 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset

1178 
val imp_abs_neg_id = thm "imp_abs_neg_id"; *) 
14738  1179 
val abs_leI = thm "abs_leI"; 
1180 
val le_minus_self_iff = thm "le_minus_self_iff"; 

1181 
val minus_le_self_iff = thm "minus_le_self_iff"; 

1182 
val abs_le_D1 = thm "abs_le_D1"; 

1183 
val abs_le_D2 = thm "abs_le_D2"; 

1184 
val abs_le_iff = thm "abs_le_iff"; 

1185 
val abs_triangle_ineq = thm "abs_triangle_ineq"; 

1186 
val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq"; 

1187 
*} 

1188 

1189 
end 