1 (* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
3 header {* Complete lattices *}
5 theory Complete_Lattices
10 less_eq (infix "\<sqsubseteq>" 50) and
11 less (infix "\<sqsubset>" 50)
14 subsection {* Syntactic infimum and supremum operations *}
17 fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
20 fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
23 subsection {* Abstract complete lattices *}
25 text {* A complete lattice always has a bottom and a top,
26 so we include them into the following type class,
27 along with assumptions that define bottom and top
28 in terms of infimum and supremum. *}
30 class complete_lattice = lattice + Inf + Sup + bot + top +
31 assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
32 and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
33 assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
34 and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
35 assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
36 assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
39 subclass bounded_lattice
42 show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
43 show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
46 lemma dual_complete_lattice:
47 "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
48 by (auto intro!: class.complete_lattice.intro dual_lattice)
49 (unfold_locales, (fact Inf_empty Sup_empty
50 Sup_upper Sup_least Inf_lower Inf_greatest)+)
52 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
53 INF_def: "INFI A f = \<Sqinter>(f ` A)"
55 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
56 SUP_def: "SUPR A f = \<Squnion>(f ` A)"
59 Note: must use names @{const INFI} and @{const SUPR} here instead of
60 @{text INF} and @{text SUP} to allow the following syntax coexist
61 with the plain constant names.
67 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
68 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
69 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10)
70 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10)
73 "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
74 "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
75 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
76 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
79 "INF x y. B" == "INF x. INF y. B"
80 "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
81 "INF x. B" == "INF x:CONST UNIV. B"
82 "INF x:A. B" == "CONST INFI A (%x. B)"
83 "SUP x y. B" == "SUP x. SUP y. B"
84 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
85 "SUP x. B" == "SUP x:CONST UNIV. B"
86 "SUP x:A. B" == "CONST SUPR A (%x. B)"
89 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
90 Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
91 *} -- {* to avoid eta-contraction of body *}
93 context complete_lattice
96 lemma INF_foundation_dual [no_atp]:
97 "complete_lattice.SUPR Inf = INFI"
98 by (simp add: fun_eq_iff INF_def
99 complete_lattice.SUP_def [OF dual_complete_lattice])
101 lemma SUP_foundation_dual [no_atp]:
102 "complete_lattice.INFI Sup = SUPR"
103 by (simp add: fun_eq_iff SUP_def
104 complete_lattice.INF_def [OF dual_complete_lattice])
107 "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x"
108 by (blast intro: antisym Sup_least Sup_upper)
111 "(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x"
112 by (blast intro: antisym Inf_greatest Inf_lower)
115 "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x"
116 unfolding SUP_def by (rule Sup_eqI) auto
119 "(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x"
120 unfolding INF_def by (rule Inf_eqI) auto
122 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
123 by (auto simp add: INF_def intro: Inf_lower)
125 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
126 by (auto simp add: INF_def intro: Inf_greatest)
128 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
129 by (auto simp add: SUP_def intro: Sup_upper)
131 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
132 by (auto simp add: SUP_def intro: Sup_least)
134 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
135 using Inf_lower [of u A] by auto
137 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
138 using INF_lower [of i A f] by auto
140 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
141 using Sup_upper [of u A] by auto
143 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
144 using SUP_upper [of i A f] by auto
146 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
147 by (auto intro: Inf_greatest dest: Inf_lower)
149 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
150 by (auto simp add: INF_def le_Inf_iff)
152 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
153 by (auto intro: Sup_least dest: Sup_upper)
155 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
156 by (auto simp add: SUP_def Sup_le_iff)
158 lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
159 by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
161 lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
162 by (simp add: INF_def)
164 lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
165 by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
167 lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
168 by (simp add: SUP_def)
170 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
171 by (simp add: INF_def)
173 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
174 by (simp add: SUP_def)
176 lemma Inf_UNIV [simp]:
177 "\<Sqinter>UNIV = \<bottom>"
178 by (auto intro!: antisym Inf_lower)
180 lemma Sup_UNIV [simp]:
181 "\<Squnion>UNIV = \<top>"
182 by (auto intro!: antisym Sup_upper)
184 lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
185 by (simp add: INF_def image_image)
187 lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
188 by (simp add: SUP_def image_image)
190 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
191 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
193 lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
194 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
196 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
197 by (auto intro: Inf_greatest Inf_lower)
199 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
200 by (auto intro: Sup_least Sup_upper)
203 "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)"
204 by (simp add: INF_def image_def)
207 "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)"
208 by (simp add: SUP_def image_def)
211 assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
212 shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
213 proof (rule Inf_greatest)
214 fix b assume "b \<in> B"
215 with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
216 from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
217 with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
221 "(\<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)"
222 unfolding INF_def by (rule Inf_mono) fast
225 assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
226 shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
227 proof (rule Sup_least)
228 fix a assume "a \<in> A"
229 with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
230 from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
231 with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
235 "(\<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)"
236 unfolding SUP_def by (rule Sup_mono) fast
238 lemma INF_superset_mono:
239 "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
240 -- {* The last inclusion is POSITIVE! *}
241 by (blast intro: INF_mono dest: subsetD)
243 lemma SUP_subset_mono:
244 "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
245 by (blast intro: SUP_mono dest: subsetD)
248 assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
250 shows "\<Sqinter>A \<sqsubseteq> u"
252 from `A \<noteq> {}` obtain v where "v \<in> A" by blast
253 moreover with assms have "v \<sqsubseteq> u" by blast
254 ultimately show ?thesis by (rule Inf_lower2)
258 assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
260 shows "u \<sqsubseteq> \<Squnion>A"
262 from `A \<noteq> {}` obtain v where "v \<in> A" by blast
263 moreover with assms have "u \<sqsubseteq> v" by blast
264 ultimately show ?thesis by (rule Sup_upper2)
268 assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
269 assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
270 shows "(SUP i:A. f i) = (SUP j:B. g j)"
271 by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
274 assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
275 assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
276 shows "(INF i:A. f i) = (INF j:B. g j)"
277 by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
279 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
280 by (auto intro: Inf_greatest Inf_lower)
282 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
283 by (auto intro: Sup_least Sup_upper)
285 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
286 by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
289 "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
290 by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
292 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
293 by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
296 "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
297 by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
299 lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
300 by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
302 lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R")
304 show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
306 show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
309 lemma Inf_top_conv [simp, no_atp]:
310 "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
311 "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
313 show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
315 assume "\<forall>x\<in>A. x = \<top>"
316 then have "A = {} \<or> A = {\<top>}" by auto
317 then show "\<Sqinter>A = \<top>" by auto
319 assume "\<Sqinter>A = \<top>"
320 show "\<forall>x\<in>A. x = \<top>"
322 assume "\<not> (\<forall>x\<in>A. x = \<top>)"
323 then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
324 then obtain B where "A = insert x B" by blast
325 with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
328 then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
331 lemma INF_top_conv [simp]:
332 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
333 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
334 by (auto simp add: INF_def)
336 lemma Sup_bot_conv [simp, no_atp]:
337 "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
338 "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
339 using dual_complete_lattice
340 by (rule complete_lattice.Inf_top_conv)+
342 lemma SUP_bot_conv [simp]:
343 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
344 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
345 by (auto simp add: SUP_def)
347 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
348 by (auto intro: antisym INF_lower INF_greatest)
350 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
351 by (auto intro: antisym SUP_upper SUP_least)
353 lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
354 by (cases "A = {}") simp_all
356 lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
357 by (cases "A = {}") simp_all
359 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)"
360 by (iprover intro: INF_lower INF_greatest order_trans antisym)
362 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)"
363 by (iprover intro: SUP_upper SUP_least order_trans antisym)
367 shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
369 from assms obtain J where "I = insert k J" by blast
370 then show ?thesis by (simp add: INF_insert)
375 shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
377 from assms obtain J where "I = insert k J" by blast
378 then show ?thesis by (simp add: SUP_insert)
382 "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
386 "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
390 assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
392 note `y < (\<Sqinter>i\<in>A. f i)`
393 also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
395 finally show "y < f i" .
399 assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
401 have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
403 also note `(\<Squnion>i\<in>A. f i) < y`
404 finally show "f i < y" .
407 lemma INF_UNIV_bool_expand:
408 "(\<Sqinter>b. A b) = A True \<sqinter> A False"
409 by (simp add: UNIV_bool INF_insert inf_commute)
411 lemma SUP_UNIV_bool_expand:
412 "(\<Squnion>b. A b) = A True \<squnion> A False"
413 by (simp add: UNIV_bool SUP_insert sup_commute)
415 lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
416 by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
418 lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f"
419 unfolding INF_def SUP_def by (rule Inf_le_Sup) auto
423 class complete_distrib_lattice = complete_lattice +
424 assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
425 assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
429 "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
430 by (simp add: INF_def sup_Inf image_image)
433 "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
434 by (simp add: SUP_def inf_Sup image_image)
436 lemma dual_complete_distrib_lattice:
437 "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
438 apply (rule class.complete_distrib_lattice.intro)
439 apply (fact dual_complete_lattice)
440 apply (rule class.complete_distrib_lattice_axioms.intro)
441 apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
444 subclass distrib_lattice proof
446 from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
447 then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
451 "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
452 by (simp add: sup_Inf sup_commute)
455 "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
456 by (simp add: inf_Sup inf_commute)
459 "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
460 by (simp add: sup_INF sup_commute)
463 "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
464 by (simp add: inf_SUP inf_commute)
466 lemma Inf_sup_eq_top_iff:
467 "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
468 by (simp only: Inf_sup INF_top_conv)
470 lemma Sup_inf_eq_bot_iff:
471 "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
472 by (simp only: Sup_inf SUP_bot_conv)
474 lemma INF_sup_distrib2:
475 "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
476 by (subst INF_commute) (simp add: sup_INF INF_sup)
478 lemma SUP_inf_distrib2:
479 "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
480 by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
484 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
487 lemma dual_complete_boolean_algebra:
488 "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
489 by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
492 "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
494 show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
495 by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
496 show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
497 by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
500 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
501 by (simp add: INF_def SUP_def uminus_Inf image_image)
504 "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
506 have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
507 then show ?thesis by simp
510 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
511 by (simp add: INF_def SUP_def uminus_Sup image_image)
515 class complete_linorder = linorder + complete_lattice
518 lemma dual_complete_linorder:
519 "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
520 by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
522 lemma complete_linorder_inf_min: "inf = min"
523 by (auto intro: antisym simp add: min_def fun_eq_iff)
525 lemma complete_linorder_sup_max: "sup = max"
526 by (auto intro: antisym simp add: max_def fun_eq_iff)
529 "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
530 unfolding not_le [symmetric] le_Inf_iff by auto
533 "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
534 unfolding INF_def Inf_less_iff by auto
537 "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
538 unfolding not_le [symmetric] Sup_le_iff by auto
541 "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
542 unfolding SUP_def less_Sup_iff by auto
544 lemma Sup_eq_top_iff [simp]:
545 "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
547 assume *: "\<Squnion>A = \<top>"
548 show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
549 proof (intro allI impI)
550 fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
551 unfolding less_Sup_iff by auto
554 assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
555 show "\<Squnion>A = \<top>"
557 assume "\<Squnion>A \<noteq> \<top>"
558 with top_greatest [of "\<Squnion>A"]
559 have "\<Squnion>A < \<top>" unfolding le_less by auto
560 then have "\<Squnion>A < \<Squnion>A"
561 using * unfolding less_Sup_iff by auto
562 then show False by auto
566 lemma SUP_eq_top_iff [simp]:
567 "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
568 unfolding SUP_def by auto
570 lemma Inf_eq_bot_iff [simp]:
571 "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
572 using dual_complete_linorder
573 by (rule complete_linorder.Sup_eq_top_iff)
575 lemma INF_eq_bot_iff [simp]:
576 "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
577 unfolding INF_def by auto
579 lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
581 fix y assume "x \<le> \<Squnion>A" "y < x"
582 then have "y < \<Squnion>A" by auto
583 then show "\<exists>a\<in>A. y < a"
584 unfolding less_Sup_iff .
585 qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
587 lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
588 unfolding le_Sup_iff SUP_def by simp
590 lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
592 fix y assume "x \<ge> \<Sqinter>A" "y > x"
593 then have "y > \<Sqinter>A" by auto
594 then show "\<exists>a\<in>A. y > a"
595 unfolding Inf_less_iff .
596 qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
599 "INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
600 unfolding Inf_le_iff INF_def by simp
602 subclass complete_distrib_lattice
605 show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
606 by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
607 (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
608 le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
614 subsection {* Complete lattice on @{typ bool} *}
616 instantiation bool :: complete_lattice
620 [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
623 [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
626 qed (auto intro: bool_induct)
630 lemma not_False_in_image_Ball [simp]:
631 "False \<notin> P ` A \<longleftrightarrow> Ball A P"
634 lemma True_in_image_Bex [simp]:
635 "True \<in> P ` A \<longleftrightarrow> Bex A P"
638 lemma INF_bool_eq [simp]:
640 by (simp add: fun_eq_iff INF_def)
642 lemma SUP_bool_eq [simp]:
644 by (simp add: fun_eq_iff SUP_def)
646 instance bool :: complete_boolean_algebra proof
647 qed (auto intro: bool_induct)
650 subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
652 instantiation "fun" :: (type, complete_lattice) complete_lattice
656 "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
658 lemma Inf_apply [simp, code]:
659 "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
660 by (simp add: Inf_fun_def)
663 "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
665 lemma Sup_apply [simp, code]:
666 "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
667 by (simp add: Sup_fun_def)
670 qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
674 lemma INF_apply [simp]:
675 "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
676 by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def)
678 lemma SUP_apply [simp]:
679 "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
680 by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def)
682 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
683 qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image)
685 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
688 subsection {* Complete lattice on unary and binary predicates *}
690 lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
693 lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
696 lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
699 lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
702 lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
705 lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
708 lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
711 lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
714 lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
717 lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
720 lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
723 lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
726 lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
729 lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
733 subsection {* Complete lattice on @{typ "_ set"} *}
735 instantiation "set" :: (type) complete_lattice
739 "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
742 "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
745 qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
749 instance "set" :: (type) complete_boolean_algebra
751 qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
754 subsubsection {* Inter *}
756 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
757 "Inter S \<equiv> \<Sqinter>S"
760 Inter ("\<Inter>_" [900] 900)
763 "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
766 have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
768 then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
769 by (simp add: Inf_set_def image_def)
772 lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
773 by (unfold Inter_eq) blast
775 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
776 by (simp add: Inter_eq)
779 \medskip A ``destruct'' rule -- every @{term X} in @{term C}
780 contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
781 @{prop "X \<in> C"} does not! This rule is analogous to @{text spec}.
784 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
787 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
788 -- {* ``Classical'' elimination rule -- does not require proving
789 @{prop "X \<in> C"}. *}
790 by (unfold Inter_eq) blast
792 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
796 "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
797 by (fact Inf_less_eq)
799 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
800 by (fact Inf_greatest)
802 lemma Inter_empty: "\<Inter>{} = UNIV"
803 by (fact Inf_empty) (* already simp *)
805 lemma Inter_UNIV: "\<Inter>UNIV = {}"
806 by (fact Inf_UNIV) (* already simp *)
808 lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
809 by (fact Inf_insert) (* already simp *)
811 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
812 by (fact less_eq_Inf_inter)
814 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
815 by (fact Inf_union_distrib)
817 lemma Inter_UNIV_conv [simp, no_atp]:
818 "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
819 "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
820 by (fact Inf_top_conv)+
822 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
823 by (fact Inf_superset_mono)
826 subsubsection {* Intersections of families *}
828 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
829 "INTER \<equiv> INFI"
832 Note: must use name @{const INTER} here instead of @{text INT}
833 to allow the following syntax coexist with the plain constant name.
837 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
838 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
841 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
842 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
844 syntax (latex output)
845 "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
846 "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
849 "INT x y. B" == "INT x. INT y. B"
850 "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
851 "INT x. B" == "INT x:CONST UNIV. B"
852 "INT x:A. B" == "CONST INTER A (%x. B)"
855 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
856 *} -- {* to avoid eta-contraction of body *}
859 "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
860 by (auto simp add: INF_def)
862 lemma Inter_image_eq [simp]:
863 "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
864 by (rule sym) (fact INF_def)
866 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
867 by (auto simp add: INF_def image_def)
869 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
870 by (auto simp add: INF_def image_def)
872 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
875 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"
876 -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
877 by (auto simp add: INF_def image_def)
879 lemma INT_cong [cong]:
880 "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)"
883 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
886 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
889 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
892 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
893 by (fact INF_greatest)
895 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
898 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
901 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
904 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
907 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
910 lemma INT_insert_distrib:
911 "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
914 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
915 by (fact INF_constant)
917 lemma INTER_UNIV_conv:
918 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
919 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
920 by (fact INF_top_conv)+ (* already simp *)
922 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
923 by (fact INF_UNIV_bool_expand)
926 "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
927 -- {* The last inclusion is POSITIVE! *}
928 by (fact INF_superset_mono)
930 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
933 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
937 subsubsection {* Union *}
939 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
940 "Union S \<equiv> \<Squnion>S"
943 Union ("\<Union>_" [900] 900)
946 "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
949 have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
951 then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
952 by (simp add: Sup_set_def image_def)
955 lemma Union_iff [simp, no_atp]:
956 "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
957 by (unfold Union_eq) blast
959 lemma UnionI [intro]:
960 "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
961 -- {* The order of the premises presupposes that @{term C} is rigid;
962 @{term A} may be flexible. *}
965 lemma UnionE [elim!]:
966 "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
969 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
972 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
975 lemma Union_empty: "\<Union>{} = {}"
976 by (fact Sup_empty) (* already simp *)
978 lemma Union_UNIV: "\<Union>UNIV = UNIV"
979 by (fact Sup_UNIV) (* already simp *)
981 lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
982 by (fact Sup_insert) (* already simp *)
984 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
985 by (fact Sup_union_distrib)
987 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
988 by (fact Sup_inter_less_eq)
990 lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
991 by (fact Sup_bot_conv) (* already simp *)
993 lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
994 by (fact Sup_bot_conv) (* already simp *)
996 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
999 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
1002 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
1003 by (fact Sup_subset_mono)
1006 subsubsection {* Unions of families *}
1008 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
1009 "UNION \<equiv> SUPR"
1012 Note: must use name @{const UNION} here instead of @{text UN}
1013 to allow the following syntax coexist with the plain constant name.
1017 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
1018 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10)
1021 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
1022 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
1024 syntax (latex output)
1025 "_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
1026 "_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
1029 "UN x y. B" == "UN x. UN y. B"
1030 "UN x. B" == "CONST UNION CONST UNIV (%x. B)"
1031 "UN x. B" == "UN x:CONST UNIV. B"
1032 "UN x:A. B" == "CONST UNION A (%x. B)"
1035 Note the difference between ordinary xsymbol syntax of indexed
1036 unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
1037 and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
1038 former does not make the index expression a subscript of the
1039 union/intersection symbol because this leads to problems with nested
1040 subscripts in Proof General.
1043 print_translation {*
1044 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
1045 *} -- {* to avoid eta-contraction of body *}
1047 lemma UNION_eq [no_atp]:
1048 "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
1049 by (auto simp add: SUP_def)
1051 lemma bind_UNION [code]:
1052 "Set.bind A f = UNION A f"
1053 by (simp add: bind_def UNION_eq)
1055 lemma member_bind [simp]:
1056 "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
1057 by (simp add: bind_UNION)
1059 lemma Union_image_eq [simp]:
1060 "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
1061 by (rule sym) (fact SUP_def)
1063 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
1064 by (auto simp add: SUP_def image_def)
1066 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
1067 -- {* The order of the premises presupposes that @{term A} is rigid;
1068 @{term b} may be flexible. *}
1071 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"
1072 by (auto simp add: SUP_def image_def)
1074 lemma UN_cong [cong]:
1075 "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)"
1078 lemma strong_UN_cong:
1079 "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)"
1080 by (unfold simp_implies_def) (fact UN_cong)
1082 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
1085 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
1088 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
1091 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
1094 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
1097 lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
1100 lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
1101 by (fact SUP_bot) (* already simp *)
1103 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
1104 by (fact SUP_absorb)
1106 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
1107 by (fact SUP_insert)
1109 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)"
1112 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)"
1115 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
1116 by (fact SUP_le_iff)
1118 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
1119 by (fact SUP_constant)
1121 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
1124 lemma UNION_empty_conv:
1125 "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
1126 "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
1127 by (fact SUP_bot_conv)+ (* already simp *)
1129 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
1132 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
1135 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
1138 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
1139 by (auto simp add: split_if_mem2)
1141 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
1142 by (fact SUP_UNIV_bool_expand)
1144 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
1148 "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
1149 (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
1150 by (fact SUP_subset_mono)
1152 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
1155 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
1158 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
1159 -- {* NOT suitable for rewriting *}
1162 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
1165 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
1169 subsubsection {* Distributive laws *}
1171 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
1174 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
1177 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
1180 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)"
1181 by (rule sym) (rule INF_inf_distrib)
1183 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)"
1184 by (rule sym) (rule SUP_sup_distrib)
1186 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
1187 by (simp only: INT_Int_distrib INF_def)
1189 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
1190 -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
1191 -- {* Union of a family of unions *}
1192 by (simp only: UN_Un_distrib SUP_def)
1194 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
1197 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
1198 -- {* Halmos, Naive Set Theory, page 35. *}
1201 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)"
1202 by (fact SUP_inf_distrib2)
1204 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)"
1205 by (fact INF_sup_distrib2)
1207 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
1208 by (fact Sup_inf_eq_bot_iff)
1211 subsubsection {* Complement *}
1213 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
1214 by (fact uminus_INF)
1216 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
1217 by (fact uminus_SUP)
1220 subsubsection {* Miniscoping and maxiscoping *}
1222 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
1223 and Intersections. *}
1225 lemma UN_simps [simp]:
1226 "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
1227 "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
1228 "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
1229 "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
1230 "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
1231 "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
1232 "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
1233 "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
1234 "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
1235 "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
1238 lemma INT_simps [simp]:
1239 "\<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)"
1240 "\<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))"
1241 "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
1242 "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
1243 "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
1244 "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
1245 "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
1246 "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
1247 "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
1248 "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
1251 lemma UN_ball_bex_simps [simp, no_atp]:
1252 "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
1253 "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
1254 "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
1255 "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
1259 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
1261 lemma UN_extend_simps:
1262 "\<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)))"
1263 "\<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))"
1264 "\<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))"
1265 "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
1266 "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
1267 "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
1268 "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
1269 "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
1270 "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
1271 "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
1274 lemma INT_extend_simps:
1275 "\<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))"
1276 "\<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))"
1277 "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
1278 "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
1279 "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
1280 "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
1281 "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
1282 "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
1283 "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
1284 "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
1290 less_eq (infix "\<sqsubseteq>" 50) and
1291 less (infix "\<sqsubset>" 50)
1294 insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
1295 mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
1296 -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}