moving UNIV = ... equations to their proper theories
1 (* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
3 header {* Complete lattices, with special focus on sets *}
5 theory Complete_Lattice
10 less_eq (infix "\<sqsubseteq>" 50) and
11 less (infix "\<sqsubset>" 50) and
12 inf (infixl "\<sqinter>" 70) and
13 sup (infixl "\<squnion>" 65) and
18 subsection {* Syntactic infimum and supremum operations *}
21 fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
24 fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
26 subsection {* Abstract complete lattices *}
28 class complete_lattice = bounded_lattice + Inf + Sup +
29 assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
30 and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
31 assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
32 and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
35 lemma dual_complete_lattice:
36 "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
37 by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
38 (unfold_locales, (fact bot_least top_greatest
39 Sup_upper Sup_least Inf_lower Inf_greatest)+)
41 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
42 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
44 lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
45 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
47 lemma Inf_empty [simp]:
48 "\<Sqinter>{} = \<top>"
49 by (auto intro: antisym Inf_greatest)
51 lemma Sup_empty [simp]:
52 "\<Squnion>{} = \<bottom>"
53 by (auto intro: antisym Sup_least)
55 lemma Inf_UNIV [simp]:
56 "\<Sqinter>UNIV = \<bottom>"
57 by (simp add: Sup_Inf Sup_empty [symmetric])
59 lemma Sup_UNIV [simp]:
60 "\<Squnion>UNIV = \<top>"
61 by (simp add: Inf_Sup Inf_empty [symmetric])
63 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
64 by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
66 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
67 by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
69 lemma Inf_singleton [simp]:
71 by (auto intro: antisym Inf_lower Inf_greatest)
73 lemma Sup_singleton [simp]:
75 by (auto intro: antisym Sup_upper Sup_least)
78 "\<Sqinter>{a, b} = a \<sqinter> b"
79 by (simp add: Inf_insert)
82 "\<Squnion>{a, b} = a \<squnion> b"
83 by (simp add: Sup_insert)
85 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
86 by (auto intro: Inf_greatest dest: Inf_lower)
88 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
89 by (auto intro: Sup_least dest: Sup_upper)
92 assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
93 shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
94 proof (rule Inf_greatest)
95 fix b assume "b \<in> B"
96 with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
97 from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
98 with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
102 assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
103 shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
104 proof (rule Sup_least)
105 fix a assume "a \<in> A"
106 with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
107 from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
108 with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
111 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
112 using Sup_upper[of u A] by auto
114 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
115 using Inf_lower[of u A] by auto
117 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
118 "INFI A f = \<Sqinter> (f ` A)"
120 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
121 "SUPR A f = \<Squnion> (f ` A)"
126 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
127 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
128 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10)
129 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10)
132 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
133 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
134 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
135 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
138 "INF x y. B" == "INF x. INF y. B"
139 "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
140 "INF x. B" == "INF x:CONST UNIV. B"
141 "INF x:A. B" == "CONST INFI A (%x. B)"
142 "SUP x y. B" == "SUP x. SUP y. B"
143 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
144 "SUP x. B" == "SUP x:CONST UNIV. B"
145 "SUP x:A. B" == "CONST SUPR A (%x. B)"
148 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
149 Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
150 *} -- {* to avoid eta-contraction of body *}
152 context complete_lattice
155 lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"
156 by (simp add: INFI_def)
158 lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
159 by (simp add: INFI_def Inf_insert)
161 lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
162 by (auto simp add: INFI_def intro: Inf_lower)
164 lemma INF_leI2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
165 using INF_leI [of i A f] by auto
167 lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
168 by (auto simp add: INFI_def intro: Inf_greatest)
170 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> f i)"
171 by (auto simp add: INFI_def le_Inf_iff)
173 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
174 by (auto intro: antisym INF_leI le_INFI)
177 "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
178 by (simp add: INFI_def image_def)
181 "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
182 by (force intro!: Inf_mono simp: INFI_def)
184 lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
185 by (intro INF_mono) auto
187 lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
188 by (iprover intro: INF_leI le_INFI order_trans antisym)
191 "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
192 by (simp add: SUPR_def image_def)
194 lemma le_SUPI: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
195 by (auto simp add: SUPR_def intro: Sup_upper)
197 lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
198 using le_SUPI [of i A f] by auto
200 lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
201 by (auto simp add: SUPR_def intro: Sup_least)
203 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. f i \<sqsubseteq> u)"
204 unfolding SUPR_def by (auto simp add: Sup_le_iff)
206 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
207 by (auto intro: antisym SUP_leI le_SUPI)
210 "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
211 by (force intro!: Sup_mono simp: SUPR_def)
213 lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
214 by (intro SUP_mono) auto
216 lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
217 by (iprover intro: SUP_leI le_SUPI order_trans antisym)
219 lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
220 by (simp add: SUPR_def)
222 lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
223 by (simp add: SUPR_def Sup_insert)
228 fixes a :: "'a\<Colon>{complete_lattice,linorder}"
229 shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
230 unfolding not_le [symmetric] le_Inf_iff by auto
233 fixes a :: "'a::{complete_lattice,linorder}"
234 shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
235 unfolding INFI_def Inf_less_iff by auto
238 fixes a :: "'a\<Colon>{complete_lattice,linorder}"
239 shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
240 unfolding not_le [symmetric] Sup_le_iff by auto
243 fixes a :: "'a::{complete_lattice,linorder}"
244 shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
245 unfolding SUPR_def less_Sup_iff by auto
247 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
249 instantiation bool :: complete_lattice
253 "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
256 "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
259 qed (auto simp add: Inf_bool_def Sup_bool_def)
263 lemma INFI_bool_eq [simp]:
267 fix P :: "'a \<Rightarrow> bool"
268 show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
269 by (auto simp add: Ball_def INFI_def Inf_bool_def)
272 lemma SUPR_bool_eq [simp]:
276 fix P :: "'a \<Rightarrow> bool"
277 show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
278 by (auto simp add: Bex_def SUPR_def Sup_bool_def)
281 instantiation "fun" :: (type, complete_lattice) complete_lattice
285 "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
288 "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
289 by (simp add: Inf_fun_def)
292 "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
295 "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
296 by (simp add: Sup_fun_def)
299 qed (auto simp add: le_fun_def Inf_apply Sup_apply
300 intro: Inf_lower Sup_upper Inf_greatest Sup_least)
305 "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
306 by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
309 "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
310 by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
313 subsection {* Inter *}
315 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
316 "Inter S \<equiv> \<Sqinter>S"
319 Inter ("\<Inter>_" [90] 90)
322 "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
325 have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
327 then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
328 by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
331 lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
332 by (unfold Inter_eq) blast
334 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
335 by (simp add: Inter_eq)
338 \medskip A ``destruct'' rule -- every @{term X} in @{term C}
339 contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
340 @{prop "X \<in> C"} does not! This rule is analogous to @{text spec}.
343 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
346 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
347 -- {* ``Classical'' elimination rule -- does not require proving
348 @{prop "X \<in> C"}. *}
349 by (unfold Inter_eq) blast
351 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
354 lemma (in complete_lattice) Inf_less_eq:
355 assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
357 shows "\<Sqinter>A \<sqsubseteq> u"
359 from `A \<noteq> {}` obtain v where "v \<in> A" by blast
360 moreover with assms have "v \<sqsubseteq> u" by blast
361 ultimately show ?thesis by (rule Inf_lower2)
365 "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
366 by (fact Inf_less_eq)
368 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
369 by (fact Inf_greatest)
371 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
372 by (fact Inf_binary [symmetric])
374 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
377 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
380 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
383 lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
384 by (auto intro: Inf_greatest Inf_lower)
386 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
387 by (fact Inf_inter_less)
389 lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
390 by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
392 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
393 by (fact Inf_union_distrib)
395 lemma (in complete_lattice) Inf_top_conv [no_atp]:
396 "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
397 "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
399 show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
401 assume "\<forall>x\<in>A. x = \<top>"
402 then have "A = {} \<or> A = {\<top>}" by auto
403 then show "\<Sqinter>A = \<top>" by auto
405 assume "\<Sqinter>A = \<top>"
406 show "\<forall>x\<in>A. x = \<top>"
408 assume "\<not> (\<forall>x\<in>A. x = \<top>)"
409 then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
410 then obtain B where "A = insert x B" by blast
411 with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
414 then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
417 lemma Inter_UNIV_conv [simp,no_atp]:
418 "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
419 "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
420 by (fact Inf_top_conv)+
422 lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
423 by (auto intro: Inf_greatest Inf_lower)
425 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
426 by (fact Inf_anti_mono)
429 subsection {* Intersections of families *}
431 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
432 "INTER \<equiv> INFI"
435 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
436 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
439 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
440 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
442 syntax (latex output)
443 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
444 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
447 "INT x y. B" == "INT x. INT y. B"
448 "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
449 "INT x. B" == "INT x:CONST UNIV. B"
450 "INT x:A. B" == "CONST INTER A (%x. B)"
453 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
454 *} -- {* to avoid eta-contraction of body *}
456 lemma INTER_eq_Inter_image:
457 "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
461 "\<Inter>S = (\<Inter>x\<in>S. x)"
462 by (simp add: INTER_eq_Inter_image image_def)
465 "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
466 by (auto simp add: INTER_eq_Inter_image Inter_eq)
468 lemma Inter_image_eq [simp]:
469 "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
470 by (rule sym) (fact INFI_def)
472 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
473 by (unfold INTER_def) blast
475 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
476 by (unfold INTER_def) blast
478 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
481 lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
482 -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
483 by (unfold INTER_def) blast
485 lemma INT_cong [cong]:
486 "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
489 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
492 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
495 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
498 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
501 lemma (in complete_lattice) INFI_empty:
502 "(\<Sqinter>x\<in>{}. B x) = \<top>"
503 by (simp add: INFI_def)
505 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
508 lemma (in complete_lattice) INFI_absorb:
510 shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
512 from assms obtain J where "I = insert k J" by blast
513 then show ?thesis by (simp add: INF_insert)
516 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
517 by (fact INFI_absorb)
519 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
522 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
525 lemma (in complete_lattice) INF_union:
526 "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
527 by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INFI INF_leI)
529 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
532 lemma INT_insert_distrib:
533 "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
536 -- {* continue generalization from here *}
538 lemma (in complete_lattice) INF_constant:
539 "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
540 by (simp add: INF_empty)
542 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
543 by (fact INF_constant)
545 lemma (in complete_lattice) INF_eq:
546 "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
547 by (simp add: INFI_def image_def)
549 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
550 -- {* Look: it has an \emph{existential} quantifier *}
553 lemma (in complete_lattice) INF_top_conv:
554 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
555 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
556 by (auto simp add: INFI_def Inf_top_conv)
558 lemma INTER_UNIV_conv [simp]:
559 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
560 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
561 by (fact INF_top_conv)+
563 lemma (in complete_lattice) INFI_UNIV_range:
564 "(\<Sqinter>x\<in>UNIV. f x) = \<Sqinter>range f"
565 by (simp add: INFI_def)
567 lemma (in complete_lattice) INF_bool_eq:
568 "(\<Sqinter>b. A b) = A True \<sqinter> A False"
569 by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
571 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
572 by (fact INF_bool_eq)
574 lemma (in complete_lattice) INF_anti_mono:
575 "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
576 -- {* The last inclusion is POSITIVE! *}
577 by (blast dest: subsetD)
580 "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
581 -- {* The last inclusion is POSITIVE! *}
582 by (blast dest: subsetD)
584 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
587 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
591 subsection {* Union *}
593 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
594 "Union S \<equiv> \<Squnion>S"
597 Union ("\<Union>_" [90] 90)
600 "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
603 have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
605 then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
606 by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
609 lemma Union_iff [simp, no_atp]:
610 "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
611 by (unfold Union_eq) blast
613 lemma UnionI [intro]:
614 "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
615 -- {* The order of the premises presupposes that @{term C} is rigid;
616 @{term A} may be flexible. *}
619 lemma UnionE [elim!]:
620 "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
623 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
624 by (iprover intro: subsetI UnionI)
626 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
627 by (iprover intro: subsetI elim: UnionE dest: subsetD)
629 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
632 lemma Union_empty [simp]: "\<Union>{} = {}"
635 lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
638 lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
641 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
644 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
647 lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
650 lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
653 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
656 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
659 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
662 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
666 subsection {* Unions of families *}
668 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
669 "UNION \<equiv> SUPR"
672 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
673 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10)
676 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
677 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
679 syntax (latex output)
680 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
681 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
684 "UN x y. B" == "UN x. UN y. B"
685 "UN x. B" == "CONST UNION CONST UNIV (%x. B)"
686 "UN x. B" == "UN x:CONST UNIV. B"
687 "UN x:A. B" == "CONST UNION A (%x. B)"
690 Note the difference between ordinary xsymbol syntax of indexed
691 unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
692 and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
693 former does not make the index expression a subscript of the
694 union/intersection symbol because this leads to problems with nested
695 subscripts in Proof General.
699 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
700 *} -- {* to avoid eta-contraction of body *}
702 lemma UNION_eq_Union_image:
703 "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
707 "\<Union>S = (\<Union>x\<in>S. x)"
708 by (simp add: UNION_eq_Union_image image_def)
710 lemma UNION_def [no_atp]:
711 "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
712 by (auto simp add: UNION_eq_Union_image Union_eq)
714 lemma Union_image_eq [simp]:
715 "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
716 by (rule sym) (fact UNION_eq_Union_image)
718 lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
719 by (unfold UNION_def) blast
721 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
722 -- {* The order of the premises presupposes that @{term A} is rigid;
723 @{term b} may be flexible. *}
726 lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
727 by (unfold UNION_def) blast
729 lemma UN_cong [cong]:
730 "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
731 by (simp add: UNION_def)
733 lemma strong_UN_cong:
734 "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
735 by (simp add: UNION_def simp_implies_def)
737 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
740 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
743 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
744 by (iprover intro: subsetI elim: UN_E dest: subsetD)
746 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
749 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
752 lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
755 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
758 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
761 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
764 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
767 lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
770 lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
773 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
776 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
779 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
782 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
785 lemma UNION_empty_conv[simp]:
786 "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
787 "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
790 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
793 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
796 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
799 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
800 by (auto simp add: split_if_mem2)
802 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
803 by (auto intro: bool_contrapos)
805 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
809 "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
810 (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
811 by (blast dest: subsetD)
813 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
816 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
819 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
820 -- {* NOT suitable for rewriting *}
823 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
827 subsection {* Distributive laws *}
829 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
832 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
835 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
836 -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
837 -- {* Union of a family of unions *}
840 lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
841 -- {* Equivalent version *}
844 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
847 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
850 lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
851 -- {* Equivalent version *}
854 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
855 -- {* Halmos, Naive Set Theory, page 35. *}
858 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
861 lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
864 lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
868 subsection {* Complement *}
870 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
873 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
877 subsection {* Miniscoping and maxiscoping *}
879 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
880 and Intersections. *}
882 lemma UN_simps [simp]:
883 "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
884 "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
885 "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
886 "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
887 "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
888 "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
889 "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
890 "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
891 "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
892 "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
895 lemma INT_simps [simp]:
896 "\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter>B)"
897 "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
898 "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
899 "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
900 "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
901 "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
902 "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
903 "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
904 "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
905 "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
908 lemma ball_simps [simp,no_atp]:
909 "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
910 "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
911 "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
912 "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
913 "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
914 "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
915 "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
916 "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
917 "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
918 "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
919 "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
920 "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
923 lemma bex_simps [simp,no_atp]:
924 "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
925 "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
926 "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
927 "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
928 "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
929 "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
930 "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
931 "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
932 "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
933 "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
936 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
938 lemma UN_extend_simps:
939 "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
940 "\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))"
941 "\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))"
942 "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
943 "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
944 "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
945 "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
946 "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
947 "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
948 "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
951 lemma INT_extend_simps:
952 "\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))"
953 "\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))"
954 "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
955 "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
956 "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
957 "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
958 "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
959 "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
960 "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
961 "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
966 less_eq (infix "\<sqsubseteq>" 50) and
967 less (infix "\<sqsubset>" 50) and
968 bot ("\<bottom>") and
970 inf (infixl "\<sqinter>" 70) and
971 sup (infixl "\<squnion>" 65) and
972 Inf ("\<Sqinter>_" [900] 900) and
973 Sup ("\<Squnion>_" [900] 900)
976 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
977 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
978 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
979 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
982 insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
983 mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
984 -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}