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_empty Inf_insert)
82 "\<Squnion>{a, b} = a \<squnion> b"
83 by (simp add: Sup_empty 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
112 "\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>"
113 by (rule antisym) auto
116 "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
117 by (rule antisym) auto
119 lemma not_less_bot[simp]: "\<not> (x \<sqsubset> \<bottom>)"
120 using bot_least[of x] by (auto simp: le_less)
122 lemma not_top_less[simp]: "\<not> (\<top> \<sqsubset> x)"
123 using top_greatest[of x] by (auto simp: le_less)
125 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
126 using Sup_upper[of u A] by auto
128 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
129 using Inf_lower[of u A] by auto
131 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
132 "INFI A f = \<Sqinter> (f ` A)"
134 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
135 "SUPR A f = \<Squnion> (f ` A)"
140 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
141 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
142 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10)
143 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10)
146 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
147 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
148 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
149 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
152 "INF x y. B" == "INF x. INF y. B"
153 "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
154 "INF x. B" == "INF x:CONST UNIV. B"
155 "INF x:A. B" == "CONST INFI A (%x. B)"
156 "SUP x y. B" == "SUP x. SUP y. B"
157 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
158 "SUP x. B" == "SUP x:CONST UNIV. B"
159 "SUP x:A. B" == "CONST SUPR A (%x. B)"
162 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
163 Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
164 *} -- {* to avoid eta-contraction of body *}
166 context complete_lattice
169 lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
170 by (simp add: SUPR_def cong: image_cong)
172 lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
173 by (simp add: INFI_def cong: image_cong)
175 lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
176 by (auto simp add: SUPR_def intro: Sup_upper)
178 lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
179 using le_SUPI[of i A M] by auto
181 lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"
182 by (auto simp add: SUPR_def intro: Sup_least)
184 lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"
185 by (auto simp add: INFI_def intro: Inf_lower)
187 lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"
188 using INF_leI[of i A M] by auto
190 lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"
191 by (auto simp add: INFI_def intro: Inf_greatest)
193 lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
194 unfolding SUPR_def by (auto simp add: Sup_le_iff)
196 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
197 unfolding INFI_def by (auto simp add: le_Inf_iff)
199 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"
200 by (auto intro: antisym INF_leI le_INFI)
202 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"
203 by (auto intro: antisym SUP_leI le_SUPI)
206 "(\<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)"
207 by (force intro!: Inf_mono simp: INFI_def)
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 INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
214 by (intro INF_mono) auto
216 lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
217 by (intro SUP_mono) auto
219 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)"
220 by (iprover intro: INF_leI le_INFI order_trans antisym)
222 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)"
223 by (iprover intro: SUP_leI le_SUPI order_trans antisym)
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\<Colon>{complete_lattice,linorder}"
234 shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
235 unfolding not_le [symmetric] Sup_le_iff by auto
238 fixes a :: "'a::{complete_lattice,linorder}"
239 shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
240 unfolding INFI_def Inf_less_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 le_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>)"*)
398 lemma Inter_UNIV_conv [simp,no_atp]:
399 "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
400 "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
403 lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
404 by (auto intro: Inf_greatest Inf_lower)
406 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
407 by (fact Inf_anti_mono)
410 subsection {* Intersections of families *}
412 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
413 "INTER \<equiv> INFI"
416 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
417 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
420 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
421 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
423 syntax (latex output)
424 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
425 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
428 "INT x y. B" == "INT x. INT y. B"
429 "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
430 "INT x. B" == "INT x:CONST UNIV. B"
431 "INT x:A. B" == "CONST INTER A (%x. B)"
434 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
435 *} -- {* to avoid eta-contraction of body *}
437 lemma INTER_eq_Inter_image:
438 "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
442 "\<Inter>S = (\<Inter>x\<in>S. x)"
443 by (simp add: INTER_eq_Inter_image image_def)
446 "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
447 by (auto simp add: INTER_eq_Inter_image Inter_eq)
449 lemma Inter_image_eq [simp]:
450 "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
451 by (rule sym) (fact INTER_eq_Inter_image)
453 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
454 by (unfold INTER_def) blast
456 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
457 by (unfold INTER_def) blast
459 lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
462 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
463 -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
464 by (unfold INTER_def) blast
466 lemma INT_cong [cong]:
467 "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
468 by (simp add: INTER_def)
470 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
473 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
476 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
479 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
482 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
485 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
488 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
491 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
494 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
497 lemma INT_insert_distrib:
498 "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
501 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
504 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
505 -- {* Look: it has an \emph{existential} quantifier *}
508 lemma INTER_UNIV_conv[simp]:
509 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
510 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
513 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
514 by (auto intro: bool_induct)
516 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
520 "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
521 (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
522 -- {* The last inclusion is POSITIVE! *}
523 by (blast dest: subsetD)
525 lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
529 subsection {* Union *}
531 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
532 "Union S \<equiv> \<Squnion>S"
535 Union ("\<Union>_" [90] 90)
538 "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
541 have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
543 then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
544 by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
547 lemma Union_iff [simp, no_atp]:
548 "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
549 by (unfold Union_eq) blast
551 lemma UnionI [intro]:
552 "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
553 -- {* The order of the premises presupposes that @{term C} is rigid;
554 @{term A} may be flexible. *}
557 lemma UnionE [elim!]:
558 "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
561 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
562 by (iprover intro: subsetI UnionI)
564 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
565 by (iprover intro: subsetI elim: UnionE dest: subsetD)
567 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
570 lemma Union_empty [simp]: "Union({}) = {}"
573 lemma Union_UNIV [simp]: "Union UNIV = UNIV"
576 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
579 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
582 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
585 lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
588 lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
591 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
594 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
597 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
600 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
604 subsection {* Unions of families *}
606 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
607 "UNION \<equiv> SUPR"
610 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
611 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10)
614 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
615 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
617 syntax (latex output)
618 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
619 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
622 "UN x y. B" == "UN x. UN y. B"
623 "UN x. B" == "CONST UNION CONST UNIV (%x. B)"
624 "UN x. B" == "UN x:CONST UNIV. B"
625 "UN x:A. B" == "CONST UNION A (%x. B)"
628 Note the difference between ordinary xsymbol syntax of indexed
629 unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
630 and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
631 former does not make the index expression a subscript of the
632 union/intersection symbol because this leads to problems with nested
633 subscripts in Proof General.
637 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
638 *} -- {* to avoid eta-contraction of body *}
640 lemma UNION_eq_Union_image:
641 "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
645 "\<Union>S = (\<Union>x\<in>S. x)"
646 by (simp add: UNION_eq_Union_image image_def)
648 lemma UNION_def [no_atp]:
649 "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
650 by (auto simp add: UNION_eq_Union_image Union_eq)
652 lemma Union_image_eq [simp]:
653 "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
654 by (rule sym) (fact UNION_eq_Union_image)
656 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
657 by (unfold UNION_def) blast
659 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
660 -- {* The order of the premises presupposes that @{term A} is rigid;
661 @{term b} may be flexible. *}
664 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
665 by (unfold UNION_def) blast
667 lemma UN_cong [cong]:
668 "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
669 by (simp add: UNION_def)
671 lemma strong_UN_cong:
672 "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
673 by (simp add: UNION_def simp_implies_def)
675 lemma image_eq_UN: "f`A = (UN x:A. {f x})"
678 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
681 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
682 by (iprover intro: subsetI elim: UN_E dest: subsetD)
684 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
687 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
690 lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
693 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
696 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
699 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
702 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
705 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)"
708 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)"
711 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
714 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
717 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
720 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
723 lemma UNION_empty_conv[simp]:
724 "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
725 "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
728 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
731 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
734 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
737 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
738 by (auto simp add: split_if_mem2)
740 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
741 by (auto intro: bool_contrapos)
743 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
747 "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
748 (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
749 by (blast dest: subsetD)
751 lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
754 lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
757 lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
758 -- {* NOT suitable for rewriting *}
761 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
765 subsection {* Distributive laws *}
767 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
770 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
773 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
774 -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
775 -- {* Union of a family of unions *}
778 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)"
779 -- {* Equivalent version *}
782 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
785 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
788 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)"
789 -- {* Equivalent version *}
792 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
793 -- {* Halmos, Naive Set Theory, page 35. *}
796 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
799 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)"
802 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)"
806 subsection {* Complement *}
808 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
811 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
815 subsection {* Miniscoping and maxiscoping *}
817 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
818 and Intersections. *}
820 lemma UN_simps [simp]:
821 "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
822 "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"
823 "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"
824 "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"
825 "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"
826 "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"
827 "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"
828 "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
829 "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"
830 "!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))"
833 lemma INT_simps [simp]:
834 "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
835 "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
836 "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"
837 "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"
838 "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
839 "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"
840 "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"
841 "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
842 "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
843 "!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))"
846 lemma ball_simps [simp,no_atp]:
847 "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
848 "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
849 "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
850 "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
851 "!!P. (ALL x:{}. P x) = True"
852 "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
853 "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
854 "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
855 "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
856 "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
857 "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
858 "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
861 lemma bex_simps [simp,no_atp]:
862 "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
863 "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
864 "!!P. (EX x:{}. P x) = False"
865 "!!P. (EX x:UNIV. P x) = (EX x. P x)"
866 "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
867 "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
868 "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
869 "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
870 "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
871 "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
874 lemma ball_conj_distrib:
875 "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
878 lemma bex_disj_distrib:
879 "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
883 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
885 lemma UN_extend_simps:
886 "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
887 "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"
888 "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"
889 "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
890 "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
891 "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
892 "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
893 "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
894 "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
895 "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
898 lemma INT_extend_simps:
899 "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
900 "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
901 "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"
902 "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"
903 "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
904 "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"
905 "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"
906 "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
907 "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
908 "!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)"
913 less_eq (infix "\<sqsubseteq>" 50) and
914 less (infix "\<sqsubset>" 50) and
915 bot ("\<bottom>") and
917 inf (infixl "\<sqinter>" 70) and
918 sup (infixl "\<squnion>" 65) and
919 Inf ("\<Sqinter>_" [900] 900) and
920 Sup ("\<Squnion>_" [900] 900)
923 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
924 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
925 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
926 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
929 insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
930 mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
931 -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}