author  obua 
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child 15314  55eec5c6d401 
permissions  rwrr 
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(* Title: HOL/Finite_Set.thy 
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ID: $Id$ 

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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel 

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Additions by Jeremy Avigad in Feb 2004 
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*) 
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header {* Finite sets *} 

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15131  9 
theory Finite_Set 
15140  10 
imports Divides Power Inductive 
15131  11 
begin 
12396  12 

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subsection {* Collection of finite sets *} 

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consts Finites :: "'a set set" 

13737  16 
syntax 
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finite :: "'a set => bool" 

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translations 

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"finite A" == "A : Finites" 

12396  20 

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inductive Finites 

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intros 

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emptyI [simp, intro!]: "{} : Finites" 

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insertI [simp, intro!]: "A : Finites ==> insert a A : Finites" 

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axclass finite \<subseteq> type 

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finite: "finite UNIV" 

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13737  29 
lemma ex_new_if_finite:  "does not depend on def of finite at all" 
14661  30 
assumes "\<not> finite (UNIV :: 'a set)" and "finite A" 
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shows "\<exists>a::'a. a \<notin> A" 

32 
proof  

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from prems have "A \<noteq> UNIV" by blast 

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thus ?thesis by blast 

35 
qed 

12396  36 

37 
lemma finite_induct [case_names empty insert, induct set: Finites]: 

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"finite F ==> 

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P {} ==> (!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F" 

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 {* Discharging @{text "x \<notin> F"} entails extra work. *} 

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proof  

13421  42 
assume "P {}" and 
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insert: "!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)" 

12396  44 
assume "finite F" 
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thus "P F" 

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proof induct 

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show "P {}" . 

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fix F x assume F: "finite F" and P: "P F" 

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show "P (insert x F)" 

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proof cases 

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assume "x \<in> F" 

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hence "insert x F = F" by (rule insert_absorb) 

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with P show ?thesis by (simp only:) 

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next 

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assume "x \<notin> F" 

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from F this P show ?thesis by (rule insert) 

57 
qed 

58 
qed 

59 
qed 

60 

61 
lemma finite_subset_induct [consumes 2, case_names empty insert]: 

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"finite F ==> F \<subseteq> A ==> 

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P {} ==> (!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==> 

64 
P F" 

65 
proof  

13421  66 
assume "P {}" and insert: 
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"!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)" 

12396  68 
assume "finite F" 
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thus "F \<subseteq> A ==> P F" 

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proof induct 

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show "P {}" . 

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fix F x assume "finite F" and "x \<notin> F" 

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and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A" 

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show "P (insert x F)" 

75 
proof (rule insert) 

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from i show "x \<in> A" by blast 

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from i have "F \<subseteq> A" by blast 

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with P show "P F" . 

79 
qed 

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qed 

81 
qed 

82 

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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" 

84 
 {* The union of two finite sets is finite. *} 

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by (induct set: Finites) simp_all 

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lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A" 

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 {* Every subset of a finite set is finite. *} 

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proof  

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assume "finite B" 

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thus "!!A. A \<subseteq> B ==> finite A" 

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proof induct 

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case empty 

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thus ?case by simp 

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next 

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case (insert F x A) 

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have A: "A \<subseteq> insert x F" and r: "A  {x} \<subseteq> F ==> finite (A  {x})" . 

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show "finite A" 

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proof cases 

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assume x: "x \<in> A" 

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with A have "A  {x} \<subseteq> F" by (simp add: subset_insert_iff) 

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with r have "finite (A  {x})" . 

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hence "finite (insert x (A  {x}))" .. 

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also have "insert x (A  {x}) = A" by (rule insert_Diff) 

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finally show ?thesis . 

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next 

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show "A \<subseteq> F ==> ?thesis" . 

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assume "x \<notin> A" 

109 
with A show "A \<subseteq> F" by (simp add: subset_insert_iff) 

110 
qed 

111 
qed 

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qed 

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114 
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" 

115 
by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) 

116 

117 
lemma finite_Int [simp, intro]: "finite F  finite G ==> finite (F Int G)" 

118 
 {* The converse obviously fails. *} 

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by (blast intro: finite_subset) 

120 

121 
lemma finite_insert [simp]: "finite (insert a A) = finite A" 

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apply (subst insert_is_Un) 

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apply (simp only: finite_Un, blast) 
12396  124 
done 
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15281  126 
lemma finite_Union[simp, intro]: 
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"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)" 

128 
by (induct rule:finite_induct) simp_all 

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12396  130 
lemma finite_empty_induct: 
131 
"finite A ==> 

132 
P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A  {a})) ==> P {}" 

133 
proof  

134 
assume "finite A" 

135 
and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A  {a})" 

136 
have "P (A  A)" 

137 
proof  

138 
fix c b :: "'a set" 

139 
presume c: "finite c" and b: "finite b" 

140 
and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y  {x})" 

141 
from c show "c \<subseteq> b ==> P (b  c)" 

142 
proof induct 

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case empty 

144 
from P1 show ?case by simp 

145 
next 

146 
case (insert F x) 

147 
have "P (b  F  {x})" 

148 
proof (rule P2) 

149 
from _ b show "finite (b  F)" by (rule finite_subset) blast 

150 
from insert show "x \<in> b  F" by simp 

151 
from insert show "P (b  F)" by simp 

152 
qed 

153 
also have "b  F  {x} = b  insert x F" by (rule Diff_insert [symmetric]) 

154 
finally show ?case . 

155 
qed 

156 
next 

157 
show "A \<subseteq> A" .. 

158 
qed 

159 
thus "P {}" by simp 

160 
qed 

161 

162 
lemma finite_Diff [simp]: "finite B ==> finite (B  Ba)" 

163 
by (rule Diff_subset [THEN finite_subset]) 

164 

165 
lemma finite_Diff_insert [iff]: "finite (A  insert a B) = finite (A  B)" 

166 
apply (subst Diff_insert) 

167 
apply (case_tac "a : A  B") 

168 
apply (rule finite_insert [symmetric, THEN trans]) 

14208  169 
apply (subst insert_Diff, simp_all) 
12396  170 
done 
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13825  173 
subsubsection {* Image and Inverse Image over Finite Sets *} 
174 

175 
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" 

176 
 {* The image of a finite set is finite. *} 

177 
by (induct set: Finites) simp_all 

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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" 
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apply (frule finite_imageI) 
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apply (erule finite_subset, assumption) 
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done 
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13825  184 
lemma finite_range_imageI: 
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"finite (range g) ==> finite (range (%x. f (g x)))" 

14208  186 
apply (drule finite_imageI, simp) 
13825  187 
done 
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12396  189 
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" 
190 
proof  

191 
have aux: "!!A. finite (A  {}) = finite A" by simp 

192 
fix B :: "'a set" 

193 
assume "finite B" 

194 
thus "!!A. f`A = B ==> inj_on f A ==> finite A" 

195 
apply induct 

196 
apply simp 

197 
apply (subgoal_tac "EX y:A. f y = x & F = f ` (A  {y})") 

198 
apply clarify 

199 
apply (simp (no_asm_use) add: inj_on_def) 

14208  200 
apply (blast dest!: aux [THEN iffD1], atomize) 
12396  201 
apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) 
14208  202 
apply (frule subsetD [OF equalityD2 insertI1], clarify) 
12396  203 
apply (rule_tac x = xa in bexI) 
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apply (simp_all add: inj_on_image_set_diff) 

205 
done 

206 
qed (rule refl) 

207 

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13825  209 
lemma inj_vimage_singleton: "inj f ==> f`{a} \<subseteq> {THE x. f x = a}" 
210 
 {* The inverse image of a singleton under an injective function 

211 
is included in a singleton. *} 

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apply (auto simp add: inj_on_def) 
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apply (blast intro: the_equality [symmetric]) 
13825  214 
done 
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lemma finite_vimageI: "[finite F; inj h] ==> finite (h ` F)" 

217 
 {* The inverse image of a finite set under an injective function 

218 
is finite. *} 

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apply (induct set: Finites, simp_all) 
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apply (subst vimage_insert) 
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apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) 
13825  222 
done 
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224 

12396  225 
subsubsection {* The finite UNION of finite sets *} 
226 

227 
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" 

228 
by (induct set: Finites) simp_all 

229 

230 
text {* 

231 
Strengthen RHS to 

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@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}? 
12396  233 

234 
We'd need to prove 

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@{prop "finite C ==> ALL A B. (UNION A B) <= C > finite {x. x:A & B x \<noteq> {}}"} 
12396  236 
by induction. *} 
237 

238 
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" 

239 
by (blast intro: finite_UN_I finite_subset) 

240 

241 

242 
subsubsection {* Sigma of finite sets *} 

243 

244 
lemma finite_SigmaI [simp]: 

245 
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" 

246 
by (unfold Sigma_def) (blast intro!: finite_UN_I) 

247 

248 
lemma finite_Prod_UNIV: 

249 
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" 

250 
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") 

251 
apply (erule ssubst) 

14208  252 
apply (erule finite_SigmaI, auto) 
12396  253 
done 
254 

255 
instance unit :: finite 

256 
proof 

257 
have "finite {()}" by simp 

258 
also have "{()} = UNIV" by auto 

259 
finally show "finite (UNIV :: unit set)" . 

260 
qed 

261 

262 
instance * :: (finite, finite) finite 

263 
proof 

264 
show "finite (UNIV :: ('a \<times> 'b) set)" 

265 
proof (rule finite_Prod_UNIV) 

266 
show "finite (UNIV :: 'a set)" by (rule finite) 

267 
show "finite (UNIV :: 'b set)" by (rule finite) 

268 
qed 

269 
qed 

270 

271 

272 
subsubsection {* The powerset of a finite set *} 

273 

274 
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" 

275 
proof 

276 
assume "finite (Pow A)" 

277 
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast 

278 
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp 

279 
next 

280 
assume "finite A" 

281 
thus "finite (Pow A)" 

282 
by induct (simp_all add: finite_UnI finite_imageI Pow_insert) 

283 
qed 

284 

285 
lemma finite_converse [iff]: "finite (r^1) = finite r" 

286 
apply (subgoal_tac "r^1 = (%(x,y). (y,x))`r") 

287 
apply simp 

288 
apply (rule iffI) 

289 
apply (erule finite_imageD [unfolded inj_on_def]) 

290 
apply (simp split add: split_split) 

291 
apply (erule finite_imageI) 

14208  292 
apply (simp add: converse_def image_def, auto) 
12396  293 
apply (rule bexI) 
294 
prefer 2 apply assumption 

295 
apply simp 

296 
done 

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subsubsection {* Finiteness of transitive closure *} 
300 

301 
text {* (Thanks to Sidi Ehmety) *} 

302 

303 
lemma finite_Field: "finite r ==> finite (Field r)" 

304 
 {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} 

305 
apply (induct set: Finites) 

306 
apply (auto simp add: Field_def Domain_insert Range_insert) 

307 
done 

308 

309 
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" 

310 
apply clarify 

311 
apply (erule trancl_induct) 

312 
apply (auto simp add: Field_def) 

313 
done 

314 

315 
lemma finite_trancl: "finite (r^+) = finite r" 

316 
apply auto 

317 
prefer 2 

318 
apply (rule trancl_subset_Field2 [THEN finite_subset]) 

319 
apply (rule finite_SigmaI) 

320 
prefer 3 

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apply (blast intro: r_into_trancl' finite_subset) 
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apply (auto simp add: finite_Field) 
323 
done 

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lemma finite_cartesian_product: "[ finite A; finite B ] ==> 
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finite (A <*> B)" 
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by (rule finite_SigmaI) 
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12396  329 

330 
subsection {* Finite cardinality *} 

331 

332 
text {* 

333 
This definition, although traditional, is ugly to work with: @{text 

334 
"card A == LEAST n. EX f. A = {f i  i. i < n}"}. Therefore we have 

335 
switched to an inductive one: 

336 
*} 

337 

338 
consts cardR :: "('a set \<times> nat) set" 

339 

340 
inductive cardR 

341 
intros 

342 
EmptyI: "({}, 0) : cardR" 

343 
InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR" 

344 

345 
constdefs 

346 
card :: "'a set => nat" 

347 
"card A == THE n. (A, n) : cardR" 

348 

349 
inductive_cases cardR_emptyE: "({}, n) : cardR" 

350 
inductive_cases cardR_insertE: "(insert a A,n) : cardR" 

351 

352 
lemma cardR_SucD: "(A, n) : cardR ==> a : A > (EX m. n = Suc m)" 

353 
by (induct set: cardR) simp_all 

354 

355 
lemma cardR_determ_aux1: 

356 
"(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A  {a}, n) : cardR)" 

14208  357 
apply (induct set: cardR, auto) 
358 
apply (simp add: insert_Diff_if, auto) 

12396  359 
apply (drule cardR_SucD) 
360 
apply (blast intro!: cardR.intros) 

361 
done 

362 

363 
lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR" 

364 
by (drule cardR_determ_aux1) auto 

365 

366 
lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)" 

367 
apply (induct set: cardR) 

368 
apply (safe elim!: cardR_emptyE cardR_insertE) 

369 
apply (rename_tac B b m) 

370 
apply (case_tac "a = b") 

371 
apply (subgoal_tac "A = B") 

14208  372 
prefer 2 apply (blast elim: equalityE, blast) 
12396  373 
apply (subgoal_tac "EX C. A = insert b C & B = insert a C") 
374 
prefer 2 

375 
apply (rule_tac x = "A Int B" in exI) 

376 
apply (blast elim: equalityE) 

377 
apply (frule_tac A = B in cardR_SucD) 

378 
apply (blast intro!: cardR.intros dest!: cardR_determ_aux2) 

379 
done 

380 

381 
lemma cardR_imp_finite: "(A, n) : cardR ==> finite A" 

382 
by (induct set: cardR) simp_all 

383 

384 
lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR" 

385 
by (induct set: Finites) (auto intro!: cardR.intros) 

386 

387 
lemma card_equality: "(A,n) : cardR ==> card A = n" 

388 
by (unfold card_def) (blast intro: cardR_determ) 

389 

390 
lemma card_empty [simp]: "card {} = 0" 

391 
by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE) 

392 

393 
lemma card_insert_disjoint [simp]: 

394 
"finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)" 

395 
proof  

396 
assume x: "x \<notin> A" 

397 
hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)" 

398 
apply (auto intro!: cardR.intros) 

399 
apply (rule_tac A1 = A in finite_imp_cardR [THEN exE]) 

400 
apply (force dest: cardR_imp_finite) 

401 
apply (blast intro!: cardR.intros intro: cardR_determ) 

402 
done 

403 
assume "finite A" 

404 
thus ?thesis 

405 
apply (simp add: card_def aux) 

406 
apply (rule the_equality) 

407 
apply (auto intro: finite_imp_cardR 

408 
cong: conj_cong simp: card_def [symmetric] card_equality) 

409 
done 

410 
qed 

411 

412 
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" 

413 
apply auto 

14208  414 
apply (drule_tac a = x in mk_disjoint_insert, clarify) 
415 
apply (rotate_tac 1, auto) 

12396  416 
done 
417 

418 
lemma card_insert_if: 

419 
"finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" 

420 
by (simp add: insert_absorb) 

421 

422 
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A  {x})) = card A" 

14302  423 
apply(rule_tac t = A in insert_Diff [THEN subst], assumption) 
424 
apply(simp del:insert_Diff_single) 

425 
done 

12396  426 

427 
lemma card_Diff_singleton: 

428 
"finite A ==> x: A ==> card (A  {x}) = card A  1" 

429 
by (simp add: card_Suc_Diff1 [symmetric]) 

430 

431 
lemma card_Diff_singleton_if: 

432 
"finite A ==> card (A{x}) = (if x : A then card A  1 else card A)" 

433 
by (simp add: card_Diff_singleton) 

434 

435 
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A  {x}))" 

436 
by (simp add: card_insert_if card_Suc_Diff1) 

437 

438 
lemma card_insert_le: "finite A ==> card A <= card (insert x A)" 

439 
by (simp add: card_insert_if) 

440 

441 
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" 

14208  442 
apply (induct set: Finites, simp, clarify) 
12396  443 
apply (subgoal_tac "finite A & A  {x} <= F") 
14208  444 
prefer 2 apply (blast intro: finite_subset, atomize) 
12396  445 
apply (drule_tac x = "A  {x}" in spec) 
446 
apply (simp add: card_Diff_singleton_if split add: split_if_asm) 

14208  447 
apply (case_tac "card A", auto) 
12396  448 
done 
449 

450 
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" 

451 
apply (simp add: psubset_def linorder_not_le [symmetric]) 

452 
apply (blast dest: card_seteq) 

453 
done 

454 

455 
lemma card_mono: "finite B ==> A <= B ==> card A <= card B" 

14208  456 
apply (case_tac "A = B", simp) 
12396  457 
apply (simp add: linorder_not_less [symmetric]) 
458 
apply (blast dest: card_seteq intro: order_less_imp_le) 

459 
done 

460 

461 
lemma card_Un_Int: "finite A ==> finite B 

462 
==> card A + card B = card (A Un B) + card (A Int B)" 

14208  463 
apply (induct set: Finites, simp) 
12396  464 
apply (simp add: insert_absorb Int_insert_left) 
465 
done 

466 

467 
lemma card_Un_disjoint: "finite A ==> finite B 

468 
==> A Int B = {} ==> card (A Un B) = card A + card B" 

469 
by (simp add: card_Un_Int) 

470 

471 
lemma card_Diff_subset: 

472 
"finite A ==> B <= A ==> card A  card B = card (A  B)" 

473 
apply (subgoal_tac "(A  B) Un B = A") 

474 
prefer 2 apply blast 

14331  475 
apply (rule nat_add_right_cancel [THEN iffD1]) 
12396  476 
apply (rule card_Un_disjoint [THEN subst]) 
477 
apply (erule_tac [4] ssubst) 

478 
prefer 3 apply blast 

479 
apply (simp_all add: add_commute not_less_iff_le 

480 
add_diff_inverse card_mono finite_subset) 

481 
done 

482 

483 
lemma card_Diff1_less: "finite A ==> x: A ==> card (A  {x}) < card A" 

484 
apply (rule Suc_less_SucD) 

485 
apply (simp add: card_Suc_Diff1) 

486 
done 

487 

488 
lemma card_Diff2_less: 

489 
"finite A ==> x: A ==> y: A ==> card (A  {x}  {y}) < card A" 

490 
apply (case_tac "x = y") 

491 
apply (simp add: card_Diff1_less) 

492 
apply (rule less_trans) 

493 
prefer 2 apply (auto intro!: card_Diff1_less) 

494 
done 

495 

496 
lemma card_Diff1_le: "finite A ==> card (A  {x}) <= card A" 

497 
apply (case_tac "x : A") 

498 
apply (simp_all add: card_Diff1_less less_imp_le) 

499 
done 

500 

501 
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B" 

14208  502 
by (erule psubsetI, blast) 
12396  503 

14889  504 
lemma insert_partition: 
505 
"[ x \<notin> F; \<forall>c1\<in>insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 > c1 \<inter> c2 = {}] 

506 
==> x \<inter> \<Union> F = {}" 

507 
by auto 

508 

509 
(* main cardinality theorem *) 

510 
lemma card_partition [rule_format]: 

511 
"finite C ==> 

512 
finite (\<Union> C) > 

513 
(\<forall>c\<in>C. card c = k) > 

514 
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 > c1 \<inter> c2 = {}) > 

515 
k * card(C) = card (\<Union> C)" 

516 
apply (erule finite_induct, simp) 

517 
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition 

518 
finite_subset [of _ "\<Union> (insert x F)"]) 

519 
done 

520 

12396  521 

522 
subsubsection {* Cardinality of image *} 

523 

524 
lemma card_image_le: "finite A ==> card (f ` A) <= card A" 

14208  525 
apply (induct set: Finites, simp) 
12396  526 
apply (simp add: le_SucI finite_imageI card_insert_if) 
527 
done 

528 

529 
lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A" 

15111  530 
by (induct set: Finites, simp_all) 
12396  531 

532 
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A" 

533 
by (simp add: card_seteq card_image) 

534 

15111  535 
lemma eq_card_imp_inj_on: 
536 
"[ finite A; card(f ` A) = card A ] ==> inj_on f A" 

537 
apply(induct rule:finite_induct) 

538 
apply simp 

539 
apply(frule card_image_le[where f = f]) 

540 
apply(simp add:card_insert_if split:if_splits) 

541 
done 

542 

543 
lemma inj_on_iff_eq_card: 

544 
"finite A ==> inj_on f A = (card(f ` A) = card A)" 

545 
by(blast intro: card_image eq_card_imp_inj_on) 

546 

12396  547 

548 
subsubsection {* Cardinality of the Powerset *} 

549 

550 
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) 

551 
apply (induct set: Finites) 

552 
apply (simp_all add: Pow_insert) 

14208  553 
apply (subst card_Un_disjoint, blast) 
554 
apply (blast intro: finite_imageI, blast) 

12396  555 
apply (subgoal_tac "inj_on (insert x) (Pow F)") 
556 
apply (simp add: card_image Pow_insert) 

557 
apply (unfold inj_on_def) 

558 
apply (blast elim!: equalityE) 

559 
done 

560 

561 
text {* 

562 
\medskip Relates to equivalence classes. Based on a theorem of 

563 
F. Kammüller's. The @{prop "finite C"} premise is redundant. 

564 
*} 

565 

566 
lemma dvd_partition: 

567 
"finite C ==> finite (Union C) ==> 

568 
ALL c : C. k dvd card c ==> 

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569 
(ALL c1: C. ALL c2: C. c1 \<noteq> c2 > c1 Int c2 = {}) ==> 
12396  570 
k dvd card (Union C)" 
14208  571 
apply (induct set: Finites, simp_all, clarify) 
12396  572 
apply (subst card_Un_disjoint) 
573 
apply (auto simp add: dvd_add disjoint_eq_subset_Compl) 

574 
done 

575 

576 

577 
subsection {* A fold functional for finite sets *} 

578 

579 
text {* 

580 
For @{text n} nonnegative we have @{text "fold f e {x1, ..., xn} = 

581 
f x1 (... (f xn e))"} where @{text f} is at least leftcommutative. 

582 
*} 

583 

584 
consts 

585 
foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set" 

586 

587 
inductive "foldSet f e" 

588 
intros 

589 
emptyI [intro]: "({}, e) : foldSet f e" 

590 
insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e" 

591 

592 
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e" 

593 

594 
constdefs 

595 
fold :: "('b => 'a => 'a) => 'a => 'b set => 'a" 

596 
"fold f e A == THE x. (A, x) : foldSet f e" 

597 

598 
lemma Diff1_foldSet: "(A  {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e" 

14208  599 
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto) 
12396  600 

601 
lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A" 

602 
by (induct set: foldSet) auto 

603 

604 
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e" 

605 
by (induct set: Finites) auto 

606 

607 

608 
subsubsection {* Leftcommutative operations *} 

609 

610 
locale LC = 

611 
fixes f :: "'b => 'a => 'a" (infixl "\<cdot>" 70) 

612 
assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" 

613 

614 
lemma (in LC) foldSet_determ_aux: 

615 
"ALL A x. card A < n > (A, x) : foldSet f e > 

616 
(ALL y. (A, y) : foldSet f e > y = x)" 

617 
apply (induct n) 

618 
apply (auto simp add: less_Suc_eq) 

14208  619 
apply (erule foldSet.cases, blast) 
620 
apply (erule foldSet.cases, blast, clarify) 

12396  621 
txt {* force simplification of @{text "card A < card (insert ...)"}. *} 
622 
apply (erule rev_mp) 

623 
apply (simp add: less_Suc_eq_le) 

624 
apply (rule impI) 

625 
apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb") 

626 
apply (subgoal_tac "Aa = Ab") 

14208  627 
prefer 2 apply (blast elim!: equalityE, blast) 
12396  628 
txt {* case @{prop "xa \<notin> xb"}. *} 
629 
apply (subgoal_tac "Aa  {xb} = Ab  {xa} & xb : Aa & xa : Ab") 

14208  630 
prefer 2 apply (blast elim!: equalityE, clarify) 
12396  631 
apply (subgoal_tac "Aa = insert xb Ab  {xa}") 
632 
prefer 2 apply blast 

633 
apply (subgoal_tac "card Aa <= card Ab") 

634 
prefer 2 

635 
apply (rule Suc_le_mono [THEN subst]) 

636 
apply (simp add: card_Suc_Diff1) 

637 
apply (rule_tac A1 = "Aa  {xb}" and f1 = f in finite_imp_foldSet [THEN exE]) 

638 
apply (blast intro: foldSet_imp_finite finite_Diff) 

639 
apply (frule (1) Diff1_foldSet) 

640 
apply (subgoal_tac "ya = f xb x") 

641 
prefer 2 apply (blast del: equalityCE) 

642 
apply (subgoal_tac "(Ab  {xa}, x) : foldSet f e") 

643 
prefer 2 apply simp 

644 
apply (subgoal_tac "yb = f xa x") 

645 
prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet) 

646 
apply (simp (no_asm_simp) add: left_commute) 

647 
done 

648 

649 
lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x" 

650 
by (blast intro: foldSet_determ_aux [rule_format]) 

651 

652 
lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y" 

653 
by (unfold fold_def) (blast intro: foldSet_determ) 

654 

655 
lemma fold_empty [simp]: "fold f e {} = e" 

656 
by (unfold fold_def) blast 

657 

658 
lemma (in LC) fold_insert_aux: "x \<notin> A ==> 

659 
((insert x A, v) : foldSet f e) = 

660 
(EX y. (A, y) : foldSet f e & v = f x y)" 

661 
apply auto 

662 
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) 

663 
apply (fastsimp dest: foldSet_imp_finite) 

664 
apply (blast intro: foldSet_determ) 

665 
done 

666 

667 
lemma (in LC) fold_insert: 

668 
"finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)" 

669 
apply (unfold fold_def) 

670 
apply (simp add: fold_insert_aux) 

671 
apply (rule the_equality) 

672 
apply (auto intro: finite_imp_foldSet 

673 
cong add: conj_cong simp add: fold_def [symmetric] fold_equality) 

674 
done 

675 

676 
lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)" 

14208  677 
apply (induct set: Finites, simp) 
12396  678 
apply (simp add: left_commute fold_insert) 
679 
done 

680 

681 
lemma (in LC) fold_nest_Un_Int: 

682 
"finite A ==> finite B 

683 
==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)" 

14208  684 
apply (induct set: Finites, simp) 
12396  685 
apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb) 
686 
done 

687 

688 
lemma (in LC) fold_nest_Un_disjoint: 

689 
"finite A ==> finite B ==> A Int B = {} 

690 
==> fold f e (A Un B) = fold f (fold f e B) A" 

691 
by (simp add: fold_nest_Un_Int) 

692 

693 
declare foldSet_imp_finite [simp del] 

694 
empty_foldSetE [rule del] foldSet.intros [rule del] 

695 
 {* Delete rules to do with @{text foldSet} relation. *} 

696 

697 

698 

699 
subsubsection {* Commutative monoids *} 

700 

701 
text {* 

702 
We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"} 

703 
instead of @{text "'b => 'a => 'a"}. 

704 
*} 

705 

706 
locale ACe = 

707 
fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70) 

708 
and e :: 'a 

709 
assumes ident [simp]: "x \<cdot> e = x" 

710 
and commute: "x \<cdot> y = y \<cdot> x" 

711 
and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" 

712 

713 
lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" 

714 
proof  

715 
have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute) 

716 
also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc) 

717 
also have "z \<cdot> x = x \<cdot> z" by (simp only: commute) 

718 
finally show ?thesis . 

719 
qed 

720 

12718  721 
lemmas (in ACe) AC = assoc commute left_commute 
12396  722 

12693  723 
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x" 
12396  724 
proof  
725 
have "x \<cdot> e = x" by (rule ident) 

726 
thus ?thesis by (subst commute) 

727 
qed 

728 

729 
lemma (in ACe) fold_Un_Int: 

730 
"finite A ==> finite B ==> 

731 
fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)" 

14208  732 
apply (induct set: Finites, simp) 
13400  733 
apply (simp add: AC insert_absorb Int_insert_left 
13421  734 
LC.fold_insert [OF LC.intro]) 
12396  735 
done 
736 

737 
lemma (in ACe) fold_Un_disjoint: 

738 
"finite A ==> finite B ==> A Int B = {} ==> 

739 
fold f e (A Un B) = fold f e A \<cdot> fold f e B" 

740 
by (simp add: fold_Un_Int) 

741 

742 
lemma (in ACe) fold_Un_disjoint2: 

743 
"finite A ==> finite B ==> A Int B = {} ==> 

744 
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" 

745 
proof  

746 
assume b: "finite B" 

747 
assume "finite A" 

748 
thus "A Int B = {} ==> 

749 
fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" 

750 
proof induct 

751 
case empty 

752 
thus ?case by simp 

753 
next 

754 
case (insert F x) 

13571  755 
have "fold (f o g) e (insert x F \<union> B) = fold (f o g) e (insert x (F \<union> B))" 
12396  756 
by simp 
13571  757 
also have "... = (f o g) x (fold (f o g) e (F \<union> B))" 
13400  758 
by (rule LC.fold_insert [OF LC.intro]) 
13421  759 
(insert b insert, auto simp add: left_commute) 
13571  760 
also from insert have "fold (f o g) e (F \<union> B) = 
761 
fold (f o g) e F \<cdot> fold (f o g) e B" by blast 

762 
also have "(f o g) x ... = (f o g) x (fold (f o g) e F) \<cdot> fold (f o g) e B" 

12396  763 
by (simp add: AC) 
13571  764 
also have "(f o g) x (fold (f o g) e F) = fold (f o g) e (insert x F)" 
13400  765 
by (rule LC.fold_insert [OF LC.intro, symmetric]) (insert b insert, 
14661  766 
auto simp add: left_commute) 
12396  767 
finally show ?case . 
768 
qed 

769 
qed 

770 

771 

772 
subsection {* Generalized summation over a set *} 

773 

774 
constdefs 

14738  775 
setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" 
12396  776 
"setsum f A == if finite A then fold (op + o f) 0 A else 0" 
777 

15042  778 
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is 
779 
written @{text"\<Sum>x\<in>A. e"}. *} 

780 

12396  781 
syntax 
15074  782 
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) 
12396  783 
syntax (xsymbols) 
14738  784 
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) 
14565  785 
syntax (HTML output) 
14738  786 
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) 
15074  787 

788 
translations  {* Beware of argument permutation! *} 

789 
"SUM i:A. b" == "setsum (%i. b) A" 

790 
"\<Sum>i\<in>A. b" == "setsum (%i. b) A" 

12396  791 

15042  792 
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter 
793 
@{text"\<Sum>xP. e"}. *} 

794 

795 
syntax 

15074  796 
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ / _./ _)" [0,0,10] 10) 
15042  797 
syntax (xsymbols) 
798 
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_  (_)./ _)" [0,0,10] 10) 

799 
syntax (HTML output) 

800 
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_  (_)./ _)" [0,0,10] 10) 

801 

15074  802 
translations 
803 
"SUM xP. t" => "setsum (%x. t) {x. P}" 

804 
"\<Sum>xP. t" => "setsum (%x. t) {x. P}" 

15042  805 

806 
print_translation {* 

807 
let 

808 
fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = 

809 
(if x<>y then raise Match 

810 
else let val x' = Syntax.mark_bound x 

811 
val t' = subst_bound(x',t) 

812 
val P' = subst_bound(x',P) 

813 
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end) 

814 
in 

815 
[("setsum", setsum_tr')] 

816 
end 

817 
*} 

818 

15047  819 
text{* As Jeremy Avigad notes, setprod needs the same treatment \dots *} 
12396  820 

821 
lemma setsum_empty [simp]: "setsum f {} = 0" 

822 
by (simp add: setsum_def) 

823 

824 
lemma setsum_insert [simp]: 

825 
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" 

15047  826 
by (simp add: setsum_def LC.fold_insert [OF LC.intro] add_left_commute) 
12396  827 

14944  828 
lemma setsum_reindex [rule_format]: 
829 
"finite B ==> inj_on f B > setsum h (f ` B) = setsum (h \<circ> f) B" 

15111  830 
by (rule finite_induct, auto) 
12396  831 

14944  832 
lemma setsum_reindex_id: 
833 
"finite B ==> inj_on f B ==> setsum f B = setsum id (f ` B)" 

14485  834 
by (auto simp add: setsum_reindex id_o) 
12396  835 

836 
lemma setsum_cong: 

837 
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" 

838 
apply (case_tac "finite B") 

14208  839 
prefer 2 apply (simp add: setsum_def, simp) 
12396  840 
apply (subgoal_tac "ALL C. C <= B > (ALL x:C. f x = g x) > setsum f C = setsum g C") 
841 
apply simp 

14208  842 
apply (erule finite_induct, simp) 
843 
apply (simp add: subset_insert_iff, clarify) 

12396  844 
apply (subgoal_tac "finite C") 
845 
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) 

846 
apply (subgoal_tac "C = insert x (C  {x})") 

847 
prefer 2 apply blast 

848 
apply (erule ssubst) 

849 
apply (drule spec) 

850 
apply (erule (1) notE impE) 

14302  851 
apply (simp add: Ball_def del:insert_Diff_single) 
12396  852 
done 
853 

14944  854 
lemma setsum_reindex_cong: 
855 
"[finite A; inj_on f A; B = f ` A; !!a. g a = h (f a)] 

856 
==> setsum h B = setsum g A" 

857 
by (simp add: setsum_reindex cong: setsum_cong) 

858 

14485  859 
lemma setsum_0: "setsum (%i. 0) A = 0" 
860 
apply (case_tac "finite A") 

861 
prefer 2 apply (simp add: setsum_def) 

862 
apply (erule finite_induct, auto) 

14430
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paulson
parents:
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diff
changeset

863 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
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changeset

864 

5cb24165a2e1
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paulson
parents:
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865 
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0" 
5cb24165a2e1
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paulson
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changeset

866 
apply (subgoal_tac "setsum f F = setsum (%x. 0) F") 
5cb24165a2e1
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paulson
parents:
14331
diff
changeset

867 
apply (erule ssubst, rule setsum_0) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

868 
apply (rule setsum_cong, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
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14331
diff
changeset

869 
done 
5cb24165a2e1
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paulson
parents:
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diff
changeset

870 

14485  871 
lemma card_eq_setsum: "finite A ==> card A = setsum (%x. 1) A" 
872 
 {* Could allow many @{text "card"} proofs to be simplified. *} 

873 
by (induct set: Finites) auto 

14430
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874 

14485  875 
lemma setsum_Un_Int: "finite A ==> finite B 
876 
==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" 

877 
 {* The reversed orientation looks more natural, but LOOPS as a simprule! *} 

878 
apply (induct set: Finites, simp) 

14738  879 
apply (simp add: add_ac Int_insert_left insert_absorb) 
14485  880 
done 
881 

882 
lemma setsum_Un_disjoint: "finite A ==> finite B 

883 
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" 

884 
apply (subst setsum_Un_Int [symmetric], auto) 

885 
done 

14430
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886 

14485  887 
lemma setsum_UN_disjoint: 
888 
"finite I ==> (ALL i:I. finite (A i)) ==> 

889 
(ALL i:I. ALL j:I. i \<noteq> j > A i Int A j = {}) ==> 

890 
setsum f (UNION I A) = setsum (%i. setsum f (A i)) I" 

891 
apply (induct set: Finites, simp, atomize) 

892 
apply (subgoal_tac "ALL i:F. x \<noteq> i") 

893 
prefer 2 apply blast 

894 
apply (subgoal_tac "A x Int UNION F A = {}") 

895 
prefer 2 apply blast 

896 
apply (simp add: setsum_Un_disjoint) 

897 
done 

898 

899 
lemma setsum_Union_disjoint: 

900 
"finite C ==> (ALL A:C. finite A) ==> 

901 
(ALL A:C. ALL B:C. A \<noteq> B > A Int B = {}) ==> 

902 
setsum f (Union C) = setsum (setsum f) C" 

903 
apply (frule setsum_UN_disjoint [of C id f]) 

904 
apply (unfold Union_def id_def, assumption+) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

905 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

906 

14661  907 
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> 
15074  908 
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = 
909 
(\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))" 

14485  910 
apply (subst Sigma_def) 
911 
apply (subst setsum_UN_disjoint) 

912 
apply assumption 

913 
apply (rule ballI) 

914 
apply (drule_tac x = i in bspec, assumption) 

14661  915 
apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)") 
14485  916 
apply (rule finite_surj) 
917 
apply auto 

918 
apply (rule setsum_cong, rule refl) 

919 
apply (subst setsum_UN_disjoint) 

920 
apply (erule bspec, assumption) 

921 
apply auto 

922 
done 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

923 

14485  924 
lemma setsum_cartesian_product: "finite A ==> finite B ==> 
15074  925 
(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = 
926 
(\<Sum>z\<in>A <*> B. f (fst z) (snd z))" 

14485  927 
by (erule setsum_Sigma, auto); 
928 

929 
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)" 

930 
apply (case_tac "finite A") 

931 
prefer 2 apply (simp add: setsum_def) 

932 
apply (erule finite_induct, auto) 

14738  933 
apply (simp add: add_ac) 
14485  934 
done 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

935 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

936 
subsubsection {* Properties in more restricted classes of structures *} 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

937 

14485  938 
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" 
939 
apply (case_tac "finite A") 

940 
prefer 2 apply (simp add: setsum_def) 

941 
apply (erule rev_mp) 

942 
apply (erule finite_induct, auto) 

943 
done 

944 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

945 
lemma setsum_eq_0_iff [simp]: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

946 
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

947 
by (induct set: Finites) auto 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

948 

15047  949 
lemma setsum_constant_nat: 
15074  950 
"finite A ==> (\<Sum>x\<in>A. y) = (card A) * y" 
15047  951 
 {* Generalized to any @{text comm_semiring_1_cancel} in 
952 
@{text IntDef} as @{text setsum_constant}. *} 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

953 
by (erule finite_induct, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

954 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

955 
lemma setsum_Un: "finite A ==> finite B ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

956 
(setsum f (A Un B) :: nat) = setsum f A + setsum f B  setsum f (A Int B)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

957 
 {* For the natural numbers, we have subtraction. *} 
14738  958 
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

959 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

960 
lemma setsum_Un_ring: "finite A ==> finite B ==> 
14738  961 
(setsum f (A Un B) :: 'a :: comm_ring_1) = 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

962 
setsum f A + setsum f B  setsum f (A Int B)" 
14738  963 
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

964 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

965 
lemma setsum_diff1: "(setsum f (A  {a}) :: nat) = 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

966 
(if a:A then setsum f A  f a else setsum f A)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

967 
apply (case_tac "finite A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

968 
prefer 2 apply (simp add: setsum_def) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

969 
apply (erule finite_induct) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

970 
apply (auto simp add: insert_Diff_if) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

971 
apply (drule_tac a = a in mk_disjoint_insert, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

972 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

973 

15124  974 
(* By Jeremy Siek: *) 
975 

976 
lemma setsum_diff: 

977 
assumes finB: "finite B" 

978 
shows "B \<subseteq> A \<Longrightarrow> (setsum f (A  B) :: nat) = (setsum f A)  (setsum f B)" 

979 
using finB 

980 
proof (induct) 

981 
show "setsum f (A  {}) = (setsum f A)  (setsum f {})" by simp 

982 
next 

983 
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F" 

984 
and xFinA: "insert x F \<subseteq> A" 

985 
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A  F) = setsum f A  setsum f F" 

986 
from xnotinF xFinA have xinAF: "x \<in> (A  F)" by simp 

987 
from xinAF have A: "setsum f ((A  F)  {x}) = setsum f (A  F)  f x" 

988 
by (simp add: setsum_diff1) 

989 
from xFinA have "F \<subseteq> A" by simp 

990 
with IH have "setsum f (A  F) = setsum f A  setsum f F" by simp 

991 
with A have B: "setsum f ((A  F)  {x}) = setsum f A  setsum f F  f x" 

992 
by simp 

993 
from xnotinF have "A  insert x F = (A  F)  {x}" by auto 

994 
with B have C: "setsum f (A  insert x F) = setsum f A  setsum f F  f x" 

995 
by simp 

996 
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp 

997 
with C have "setsum f (A  insert x F) = setsum f A  setsum f (insert x F)" 

998 
by simp 

999 
thus "setsum f (A  insert x F) = setsum f A  setsum f (insert x F)" by simp 

1000 
qed 

1001 

15311
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1002 
lemma setsum_mono: 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1003 
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))" 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1004 
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)" 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1005 
proof (cases "finite K") 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1006 
case True 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1007 
thus ?thesis using le 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1008 
proof (induct) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1009 
case empty 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1010 
thus ?case by simp 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1011 
next 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1012 
case insert 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1013 
thus ?case using add_mono 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1014 
by force 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1015 
qed 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1016 
next 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1017 
case False 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1018 
thus ?thesis 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1019 
by (simp add: setsum_def) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1020 
qed 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1021 

2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1022 
lemma finite_setsum_diff1: "finite A \<Longrightarrow> (setsum f (A  {a}) :: ('a::{pordered_ab_group_add})) = 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1023 
(if a:A then setsum f A  f a else setsum f A)" 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1024 
by (erule finite_induct) (auto simp add: insert_Diff_if) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1025 

2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1026 
lemma finite_setsum_diff: 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1027 
assumes le: "finite A" "B \<subseteq> A" 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1028 
shows "setsum f (A  B) = setsum f A  ((setsum f B)::('a::pordered_ab_group_add))" 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1029 
proof  
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1030 
from le have finiteB: "finite B" using finite_subset by auto 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1031 
show ?thesis using le finiteB 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1032 
proof (induct rule: Finites.induct[OF finiteB]) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1033 
case 1 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1034 
thus ?case by auto 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1035 
next 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1036 
case 2 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1037 
thus ?case using le 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1038 
apply auto 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1039 
apply (subst Diff_insert) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1040 
apply (subst finite_setsum_diff1) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1041 
apply (auto simp add: insert_absorb) 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1042 
done 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1043 
qed 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1044 
qed 
2ca1c66a6758
Added lemmas setsum_mono, finite_setsum_diff1, finite_setsum_diff
obua
parents:
15309
diff
changeset

1045 

14738  1046 
lemma setsum_negf: "finite A ==> setsum (%x.  (f x)::'a::comm_ring_1) A = 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1047 
 setsum f A" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1048 
by (induct set: Finites, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1049 

14738  1050 
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::comm_ring_1)  g x) A = 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1051 
setsum f A  setsum g A" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1052 
by (simp add: diff_minus setsum_addf setsum_negf) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1053 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1054 
lemma setsum_nonneg: "[ finite A; 
14738  1055 
\<forall>x \<in> A. (0::'a::ordered_semidom) \<le> f x ] ==> 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1056 
0 \<le> setsum f A"; 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1057 
apply (induct set: Finites, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1058 
apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1059 
apply (blast intro: add_mono) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1060 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1061 

15308  1062 
lemma setsum_nonpos: "[ finite A; 
1063 
\<forall>x \<in> A. f x \<le> (0::'a::ordered_semidom) ] ==> 

1064 
setsum f A \<le> 0"; 

1065 
apply (induct set: Finites, auto) 

1066 
apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp) 

1067 
apply (blast intro: add_mono) 

1068 
done 

1069 

15047  1070 
lemma setsum_mult: 
1071 
fixes f :: "'a => ('b::semiring_0_cancel)" 

1072 
shows "r * setsum f A = setsum (%n. r * f n) A" 

15309  1073 
proof (cases "finite A") 
1074 
case True 

1075 
thus ?thesis 

1076 
proof (induct) 

1077 
case empty thus ?case by simp 

1078 
next 

1079 
case (insert A x) thus ?case by (simp add: right_distrib) 

1080 
qed 

15047  1081 
next 
15309  1082 
case False thus ?thesis by (simp add: setsum_def) 
15047  1083 
qed 
1084 

1085 
lemma setsum_abs: 

1086 
fixes f :: "'a => ('b::lordered_ab_group_abs)" 

1087 
assumes fin: "finite A" 

1088 
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A" 

1089 
using fin 

1090 
proof (induct) 

1091 
case empty thus ?case by simp 

1092 
next 

1093 
case (insert A x) 

1094 
thus ?case by (auto intro: abs_triangle_ineq order_trans) 

1095 
qed 

1096 

1097 
lemma setsum_abs_ge_zero: 

1098 
fixes f :: "'a => ('b::lordered_ab_group_abs)" 

1099 
assumes fin: "finite A" 

1100 
shows "0 \<le> setsum (%i. abs(f i)) A" 

1101 
using fin 

1102 
proof (induct) 

1103 
case empty thus ?case by simp 

1104 
next 

1105 
case (insert A x) thus ?case by (auto intro: order_trans) 

1106 
qed 

1107 

14485  1108 
subsubsection {* Cardinality of unions and Sigma sets *} 
1109 

1110 
lemma card_UN_disjoint: 

1111 
"finite I ==> (ALL i:I. finite (A i)) ==> 

1112 
(ALL i:I. ALL j:I. i \<noteq> j > A i Int A j = {}) ==> 

1113 
card (UNION I A) = setsum (%i. card (A i)) I" 

1114 
apply (subst card_eq_setsum) 

1115 
apply (subst finite_UN, assumption+) 

15047  1116 
apply (subgoal_tac 
1117 
"setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I") 

1118 
apply (simp add: setsum_UN_disjoint) 

1119 
apply (simp add: setsum_constant_nat cong: setsum_cong) 

14485  1120 
done 
1121 

1122 
lemma card_Union_disjoint: 

1123 
"finite C ==> (ALL A:C. finite A) ==> 

1124 
(ALL A:C. ALL B:C. A \<noteq> B > A Int B = {}) ==> 

1125 
card (Union C) = setsum card C" 

1126 
apply (frule card_UN_disjoint [of C id]) 

1127 
apply (unfold Union_def id_def, assumption+) 

1128 
done 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1129 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1130 
lemma SigmaI_insert: "y \<notin> A ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1131 
(SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1132 
by auto 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1133 

14485  1134 
lemma card_cartesian_product_singleton: "finite A ==> 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1135 
card({x} <*> A) = card(A)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1136 
apply (subgoal_tac "inj_on (%y .(x,y)) A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1137 
apply (frule card_image, assumption) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1138 
apply (subgoal_tac "(Pair x ` A) = {x} <*> A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1139 
apply (auto simp add: inj_on_def) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1140 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1141 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1142 
lemma card_SigmaI [rule_format,simp]: "finite A ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1143 
(ALL a:A. finite (B a)) > 
15074  1144 
card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1145 
apply (erule finite_induct, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1146 
apply (subst SigmaI_insert, assumption) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1147 
apply (subst card_Un_disjoint) 
14485  1148 
apply (auto intro: finite_SigmaI simp add: card_cartesian_product_singleton) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1149 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1150 

15047  1151 
lemma card_cartesian_product: 
1152 
"[ finite A; finite B ] ==> card (A <*> B) = card(A) * card(B)" 

1153 
by (simp add: setsum_constant_nat) 

1154 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1155 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1156 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1157 
subsection {* Generalized product over a set *} 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1158 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1159 
constdefs 
14738  1160 
setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1161 
"setprod f A == if finite A then fold (op * o f) 1 A else 1" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1162 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1163 
syntax 
14738  1164 
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_:_. _)" [0, 51, 10] 10) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1165 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1166 
syntax (xsymbols) 
14738  1167 
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10) 
14565  1168 
syntax (HTML output) 
14738  1169 
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1170 
translations 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1171 
"\<Prod>i:A. b" == "setprod (%i. b) A"  {* Beware of argument permutation! *} 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1172 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1173 
lemma setprod_empty [simp]: "setprod f {} = 1" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1174 
by (auto simp add: setprod_def) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1175 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1176 
lemma setprod_insert [simp]: "[ finite A; a \<notin> A ] ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1177 
setprod f (insert a A) = f a * setprod f A" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1178 
by (auto simp add: setprod_def LC_def LC.fold_insert 
14738  1179 
mult_left_commute) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1180 

14748  1181 
lemma setprod_reindex [rule_format]: 
1182 
"finite B ==> inj_on f B > setprod h (f ` B) = setprod (h \<circ> f) B" 

15111  1183 
by (rule finite_induct, auto) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1184 

14485  1185 
lemma setprod_reindex_id: "finite B ==> inj_on f B ==> 
1186 
setprod f B = setprod id (f ` B)" 

1187 
by (auto simp add: setprod_reindex id_o) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1188 

14661  1189 
lemma setprod_reindex_cong: "finite A ==> inj_on f A ==> 
14485  1190 
B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A" 
1191 
by (frule setprod_reindex, assumption, simp) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1192 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1193 
lemma setprod_cong: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1194 
"A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1195 
apply (case_tac "finite B") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1196 
prefer 2 apply (simp add: setprod_def, simp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1197 
apply (subgoal_tac "ALL C. C <= B > (ALL x:C. f x = g x) > setprod f C = setprod g C") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1198 
apply simp 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1199 
apply (erule finite_induct, simp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1200 
apply (simp add: subset_insert_iff, clarify) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1201 
apply (subgoal_tac "finite C") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1202 
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1203 
apply (subgoal_tac "C = insert x (C  {x})") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1204 
prefer 2 apply blast 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1205 
apply (erule ssubst) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1206 
apply (drule spec) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1207 
apply (erule (1) notE impE) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1208 
apply (simp add: Ball_def del:insert_Diff_single) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1209 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1210 

14485  1211 
lemma setprod_1: "setprod (%i. 1) A = 1" 
1212 
apply (case_tac "finite A") 

14738  1213 
apply (erule finite_induct, auto simp add: mult_ac) 
14485  1214 
apply (simp add: setprod_def) 
1215 
done 

1216 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1217 
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1218 
apply (subgoal_tac "setprod f F = setprod (%x. 1) F") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1219 
apply (erule ssubst, rule setprod_1) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1220 
apply (rule setprod_cong, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1221 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1222 

14485  1223 
lemma setprod_Un_Int: "finite A ==> finite B 
1224 
==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B" 

1225 
apply (induct set: Finites, simp) 

14738  1226 
apply (simp add: mult_ac insert_absorb) 
1227 
apply (simp add: mult_ac Int_insert_left insert_absorb) 

14485  1228 
done 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1229 

14485  1230 
lemma setprod_Un_disjoint: "finite A ==> finite B 
1231 
==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B" 

14738  1232 
apply (subst setprod_Un_Int [symmetric], auto simp add: mult_ac) 
14485  1233 
done 
1234 

1235 
lemma setprod_UN_disjoint: 

1236 
"finite I ==> (ALL i:I. finite (A i)) ==> 

1237 
(ALL i:I. ALL j:I. i \<noteq> j > A i Int A j = {}) ==> 

1238 
setprod f (UNION I A) = setprod (%i. setprod f (A i)) I" 

1239 
apply (induct set: Finites, simp, atomize) 

1240 
apply (subgoal_tac "ALL i:F. x \<noteq> i") 

1241 
prefer 2 apply blast 

1242 
apply (subgoal_tac "A x Int UNION F A = {}") 

1243 
prefer 2 apply blast 

1244 
apply (simp add: setprod_Un_disjoint) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1245 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1246 

14485  1247 
lemma setprod_Union_disjoint: 
1248 
"finite C ==> (ALL A:C. finite A) ==> 

1249 
(ALL A:C. ALL B:C. A \<noteq> B > A Int B = {}) ==> 

1250 
setprod f (Union C) = setprod (setprod f) C" 

1251 
apply (frule setprod_UN_disjoint [of C id f]) 

1252 
apply (unfold Union_def id_def, assumption+) 

1253 
done 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1254 

14661  1255 
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==> 
1256 
(\<Prod>x:A. (\<Prod>y: B x. f x y)) = 

1257 
(\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))" 

14485  1258 
apply (subst Sigma_def) 
1259 
apply (subst setprod_UN_disjoint) 

1260 
apply assumption 

1261 
apply (rule ballI) 

1262 
apply (drule_tac x = i in bspec, assumption) 

14661  1263 
apply (subgoal_tac "(UN y:(B i). {(i, y)}) <= (%y. (i, y)) ` (B i)") 
14485  1264 
apply (rule finite_surj) 
1265 
apply auto 

1266 
apply (rule setprod_cong, rule refl) 

1267 
apply (subst setprod_UN_disjoint) 

1268 
apply (erule bspec, assumption) 

1269 
apply auto 

1270 
done 

1271 

14661  1272 
lemma setprod_cartesian_product: "finite A ==> finite B ==> 
1273 
(\<Prod>x:A. (\<Prod>y: B. f x y)) = 

1274 
(\<Prod>z:(A <*> B). f (fst z) (snd z))" 

14485  1275 
by (erule setprod_Sigma, auto) 
1276 

1277 
lemma setprod_timesf: "setprod (%x. f x * g x) A = 

1278 
(setprod f A * setprod g A)" 

1279 
apply (case_tac "finite A") 

14738  1280 
prefer 2 apply (simp add: setprod_def mult_ac) 
14485  1281 
apply (erule finite_induct, auto) 
14738  1282 
apply (simp add: mult_ac) 
14485  1283 
done 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1284 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1285 
subsubsection {* Properties in more restricted classes of structures *} 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1286 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1287 
lemma setprod_eq_1_iff [simp]: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1288 
"finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1289 
by (induct set: Finites) auto 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1290 

15004  1291 
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1292 
apply (erule finite_induct) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1293 
apply (auto simp add: power_Suc) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1294 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1295 

15004  1296 
lemma setprod_zero: 
1297 
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1298 
apply (induct set: Finites, force, clarsimp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1299 
apply (erule disjE, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1300 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1301 

15004  1302 
lemma setprod_nonneg [rule_format]: 
1303 
"(ALL x: A. (0::'a::ordered_idom) \<le> f x) > 0 \<le> setprod f A" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1304 
apply (case_tac "finite A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1305 
apply (induct set: Finites, force, clarsimp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1306 
apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1307 
apply (rule mult_mono, assumption+) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1308 
apply (auto simp add: setprod_def) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1309 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1310 

14738  1311 
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1312 
> 0 < setprod f A" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1313 
apply (case_tac "finite A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1314 
apply (induct set: Finites, force, clarsimp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1315 
apply (subgoal_tac "0 * 0 < f x * setprod f F", force) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1316 
apply (rule mult_strict_mono, assumption+) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1317 
apply (auto simp add: setprod_def) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1318 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1319 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1320 
lemma setprod_nonzero [rule_format]: 
14738  1321 
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 > x = 0  y = 0) ==> 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1322 
finite A ==> (ALL x: A. f x \<noteq> (0::'a)) > setprod f A \<noteq> 0" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1323 
apply (erule finite_induct, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1324 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1325 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1326 
lemma setprod_zero_eq: 
14738  1327 
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 > x = 0  y = 0) ==> 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1328 
finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1329 
apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1330 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1331 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1332 
lemma setprod_nonzero_field: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1333 
"finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1334 
apply (rule setprod_nonzero, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1335 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1336 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1337 
lemma setprod_zero_eq_field: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1338 
"finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1339 
apply (rule setprod_zero_eq, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1340 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1341 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1342 
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1343 
(setprod f (A Un B) :: 'a ::{field}) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1344 
= setprod f A * setprod f B / setprod f (A Int B)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1345 
apply (subst setprod_Un_Int [symmetric], auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1346 
apply (subgoal_tac "finite (A Int B)") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1347 
apply (frule setprod_nonzero_field [of "A Int B" f], assumption) 
15228  1348 
apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1349 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1350 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1351 
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1352 
(setprod f (A  {a}) :: 'a :: {field}) = 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1353 
(if a:A then setprod f A / f a else setprod f A)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1354 
apply (erule finite_induct) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1355 
apply (auto simp add: insert_Diff_if) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1356 
apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1357 
apply (erule ssubst) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1358 
apply (subst times_divide_eq_right [THEN sym]) 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15228
diff
changeset

1359 
apply (auto simp add: mult_ac times_divide_eq_right divide_self) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1360 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1361 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1362 
lemma setprod_inversef: "finite A ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1363 
ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==> 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1364 
setprod (inverse \<circ> f) A = inverse (setprod f A)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1365 
apply (erule finite_induct) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1366 
apply (simp, simp) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1367 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1368 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1369 
lemma setprod_dividef: 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1370 
"[finite A; 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1371 
\<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})] 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1372 
==> setprod (%x. f x / g x) A = setprod f A / setprod g A" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1373 
apply (subgoal_tac 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1374 
"setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A") 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1375 
apply (erule ssubst) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1376 
apply (subst divide_inverse) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1377 
apply (subst setprod_timesf) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1378 
apply (subst setprod_inversef, assumption+, rule refl) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1379 
apply (rule setprod_cong, rule refl) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1380 
apply (subst divide_inverse, auto) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1381 
done 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1382 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1383 

5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14331
diff
changeset

1384 
subsection{* Min and Max of finite linearly ordered sets *} 
13490
44bdc150211b
Added Mi and Max on sets, hid Min and Pls on numerals.
nipkow
parents:
13421
diff
changeset

1385 