author  webertj 
Fri, 19 Oct 2012 15:12:52 +0200  
changeset 49962  a8cc904a6820 
parent 49905  a81f95693c68 
child 51328  d63ec23c9125 
permissions  rwrr 
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(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) 
11979  2 

44104  3 
header {* Complete lattices *} 
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theory Complete_Lattices 
32139  6 
imports Set 
7 
begin 

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notation 
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less_eq (infix "\<sqsubseteq>" 50) and 
46691  11 
less (infix "\<sqsubset>" 50) 
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32139  13 

32879  14 
subsection {* Syntactic infimum and supremum operations *} 
15 

16 
class Inf = 

17 
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 

18 

19 
class Sup = 

20 
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 

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46691  22 

32139  23 
subsection {* Abstract complete lattices *} 
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class complete_lattice = bounded_lattice + Inf + Sup + 
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 
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assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 
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begin 
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32678  32 
lemma dual_complete_lattice: 
44845  33 
"class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
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by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) 
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(unfold_locales, (fact bot_least top_greatest 
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Sup_upper Sup_least Inf_lower Inf_greatest)+) 
32678  37 

44040  38 
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
39 
INF_def: "INFI A f = \<Sqinter>(f ` A)" 

40 

41 
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 

42 
SUP_def: "SUPR A f = \<Squnion>(f ` A)" 

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44 
text {* 

45 
Note: must use names @{const INFI} and @{const SUPR} here instead of 

46 
@{text INF} and @{text SUP} to allow the following syntax coexist 

47 
with the plain constant names. 

48 
*} 

49 

50 
end 

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52 
syntax 

53 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) 

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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) 

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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) 

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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) 

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syntax (xsymbols) 

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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 

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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 

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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

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translations 

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"INF x y. B" == "INF x. INF y. B" 

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"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 

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"INF x. B" == "INF x:CONST UNIV. B" 

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"INF x:A. B" == "CONST INFI A (%x. B)" 

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"SUP x y. B" == "SUP x. SUP y. B" 

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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 

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"SUP x. B" == "SUP x:CONST UNIV. B" 

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"SUP x:A. B" == "CONST SUPR A (%x. B)" 

73 

74 
print_translation {* 

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[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, 

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Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] 

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*}  {* to avoid etacontraction of body *} 

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context complete_lattice 

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begin 

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44040  82 
lemma INF_foundation_dual [no_atp]: 
83 
"complete_lattice.SUPR Inf = INFI" 

44921  84 
by (simp add: fun_eq_iff INF_def 
85 
complete_lattice.SUP_def [OF dual_complete_lattice]) 

44040  86 

87 
lemma SUP_foundation_dual [no_atp]: 

88 
"complete_lattice.INFI Sup = SUPR" 

44921  89 
by (simp add: fun_eq_iff SUP_def 
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complete_lattice.INF_def [OF dual_complete_lattice]) 

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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i" 
44040  93 
by (auto simp add: INF_def intro: Inf_lower) 
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)" 
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by (auto simp add: INF_def intro: Inf_greatest) 
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)" 
44040  99 
by (auto simp add: SUP_def intro: Sup_upper) 
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u" 
44040  102 
by (auto simp add: SUP_def intro: Sup_least) 
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" 

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using Inf_lower [of u A] by auto 

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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u" 
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using INF_lower [of i A f] by auto 
44040  109 

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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A" 

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using Sup_upper [of u A] by auto 

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lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)" 
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using SUP_upper [of i A f] by auto 
44040  115 

44918  116 
lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" 
44040  117 
by (auto intro: Inf_greatest dest: Inf_lower) 
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44918  119 
lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)" 
44040  120 
by (auto simp add: INF_def le_Inf_iff) 
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44918  122 
lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" 
44040  123 
by (auto intro: Sup_least dest: Sup_upper) 
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44918  125 
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)" 
44040  126 
by (auto simp add: SUP_def Sup_le_iff) 
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41080  128 
lemma Inf_empty [simp]: 
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"\<Sqinter>{} = \<top>" 
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by (auto intro: antisym Inf_greatest) 
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44067  132 
lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" 
44040  133 
by (simp add: INF_def) 
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41080  135 
lemma Sup_empty [simp]: 
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"\<Squnion>{} = \<bottom>" 
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by (auto intro: antisym Sup_least) 
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44067  139 
lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" 
44040  140 
by (simp add: SUP_def) 
141 

41080  142 
lemma Inf_UNIV [simp]: 
143 
"\<Sqinter>UNIV = \<bottom>" 

44040  144 
by (auto intro!: antisym Inf_lower) 
41080  145 

146 
lemma Sup_UNIV [simp]: 

147 
"\<Squnion>UNIV = \<top>" 

44040  148 
by (auto intro!: antisym Sup_upper) 
41080  149 

44918  150 
lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 
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by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) 
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44040  153 
lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f" 
44919  154 
by (simp add: INF_def) 
44040  155 

44918  156 
lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) 
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44040  159 
lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f" 
44919  160 
by (simp add: SUP_def) 
44040  161 

44918  162 
lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))" 
44068  163 
by (simp add: INF_def image_image) 
164 

44918  165 
lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))" 
44068  166 
by (simp add: SUP_def image_image) 
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44040  168 
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" 
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

170 

171 
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" 

172 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

173 

43899  174 
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" 
175 
by (auto intro: Inf_greatest Inf_lower) 

176 

177 
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B" 

178 
by (auto intro: Sup_least Sup_upper) 

179 

44041  180 
lemma INF_cong: 
181 
"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)" 

182 
by (simp add: INF_def image_def) 

183 

184 
lemma SUP_cong: 

185 
"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)" 

186 
by (simp add: SUP_def image_def) 

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38705  188 
lemma Inf_mono: 
41971  189 
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" 
43741  190 
shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B" 
38705  191 
proof (rule Inf_greatest) 
192 
fix b assume "b \<in> B" 

41971  193 
with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast 
43741  194 
from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) 
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with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto 

38705  196 
qed 
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44041  198 
lemma INF_mono: 
199 
"(\<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)" 

44918  200 
unfolding INF_def by (rule Inf_mono) fast 
44041  201 

41082  202 
lemma Sup_mono: 
41971  203 
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" 
43741  204 
shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" 
41082  205 
proof (rule Sup_least) 
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fix a assume "a \<in> A" 

41971  207 
with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast 
43741  208 
from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) 
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with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto 

41082  210 
qed 
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lemma SUP_mono: 
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"(\<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)" 

44918  214 
unfolding SUP_def by (rule Sup_mono) fast 
44041  215 

216 
lemma INF_superset_mono: 

217 
"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)" 

218 
 {* The last inclusion is POSITIVE! *} 

219 
by (blast intro: INF_mono dest: subsetD) 

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221 
lemma SUP_subset_mono: 

222 
"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)" 

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by (blast intro: SUP_mono dest: subsetD) 

224 

43868  225 
lemma Inf_less_eq: 
226 
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" 

227 
and "A \<noteq> {}" 

228 
shows "\<Sqinter>A \<sqsubseteq> u" 

229 
proof  

230 
from `A \<noteq> {}` obtain v where "v \<in> A" by blast 

231 
moreover with assms have "v \<sqsubseteq> u" by blast 

232 
ultimately show ?thesis by (rule Inf_lower2) 

233 
qed 

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235 
lemma less_eq_Sup: 

236 
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v" 

237 
and "A \<noteq> {}" 

238 
shows "u \<sqsubseteq> \<Squnion>A" 

239 
proof  

240 
from `A \<noteq> {}` obtain v where "v \<in> A" by blast 

241 
moreover with assms have "u \<sqsubseteq> v" by blast 

242 
ultimately show ?thesis by (rule Sup_upper2) 

243 
qed 

244 

43899  245 
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" 
43868  246 
by (auto intro: Inf_greatest Inf_lower) 
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43899  248 
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B " 
43868  249 
by (auto intro: Sup_least Sup_upper) 
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251 
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" 

252 
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) 

253 

44041  254 
lemma INF_union: 
255 
"(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" 

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by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) 
44041  257 

43868  258 
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" 
259 
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) 

260 

44041  261 
lemma SUP_union: 
262 
"(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" 

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by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) 
44041  264 

265 
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)" 

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by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) 
44041  267 

44918  268 
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") 
269 
proof (rule antisym) 

270 
show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) 

271 
next 

272 
show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) 

273 
qed 

44041  274 

44918  275 
lemma Inf_top_conv [simp, no_atp]: 
43868  276 
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 
277 
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 

278 
proof  

279 
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 

280 
proof 

281 
assume "\<forall>x\<in>A. x = \<top>" 

282 
then have "A = {} \<or> A = {\<top>}" by auto 

44919  283 
then show "\<Sqinter>A = \<top>" by auto 
43868  284 
next 
285 
assume "\<Sqinter>A = \<top>" 

286 
show "\<forall>x\<in>A. x = \<top>" 

287 
proof (rule ccontr) 

288 
assume "\<not> (\<forall>x\<in>A. x = \<top>)" 

289 
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast 

290 
then obtain B where "A = insert x B" by blast 

44919  291 
with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp 
43868  292 
qed 
293 
qed 

294 
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto 

295 
qed 

296 

44918  297 
lemma INF_top_conv [simp]: 
44041  298 
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" 
299 
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" 

44919  300 
by (auto simp add: INF_def) 
44041  301 

44918  302 
lemma Sup_bot_conv [simp, no_atp]: 
43868  303 
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P) 
304 
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q) 

44920  305 
using dual_complete_lattice 
306 
by (rule complete_lattice.Inf_top_conv)+ 

43868  307 

44918  308 
lemma SUP_bot_conv [simp]: 
44041  309 
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" 
310 
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" 

44919  311 
by (auto simp add: SUP_def) 
44041  312 

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lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" 
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314 
by (auto intro: antisym INF_lower INF_greatest) 
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315 

43870  316 
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" 
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317 
by (auto intro: antisym SUP_upper SUP_least) 
43870  318 

44918  319 
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" 
44921  320 
by (cases "A = {}") simp_all 
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44918  322 
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" 
44921  323 
by (cases "A = {}") simp_all 
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324 

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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)" 
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326 
by (iprover intro: INF_lower INF_greatest order_trans antisym) 
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43870  328 
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)" 
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329 
by (iprover intro: SUP_upper SUP_least order_trans antisym) 
43870  330 

43871  331 
lemma INF_absorb: 
43868  332 
assumes "k \<in> I" 
333 
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" 

334 
proof  

335 
from assms obtain J where "I = insert k J" by blast 

336 
then show ?thesis by (simp add: INF_insert) 

337 
qed 

338 

43871  339 
lemma SUP_absorb: 
340 
assumes "k \<in> I" 

341 
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" 

342 
proof  

343 
from assms obtain J where "I = insert k J" by blast 

344 
then show ?thesis by (simp add: SUP_insert) 

345 
qed 

346 

347 
lemma INF_constant: 

43868  348 
"(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" 
44921  349 
by simp 
43868  350 

43871  351 
lemma SUP_constant: 
352 
"(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" 

44921  353 
by simp 
43871  354 

43943  355 
lemma less_INF_D: 
356 
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i" 

357 
proof  

358 
note `y < (\<Sqinter>i\<in>A. f i)` 

359 
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A` 

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360 
by (rule INF_lower) 
43943  361 
finally show "y < f i" . 
362 
qed 

363 

364 
lemma SUP_lessD: 

365 
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y" 

366 
proof  

367 
have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A` 

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368 
by (rule SUP_upper) 
43943  369 
also note `(\<Squnion>i\<in>A. f i) < y` 
370 
finally show "f i < y" . 

371 
qed 

372 

43873  373 
lemma INF_UNIV_bool_expand: 
43868  374 
"(\<Sqinter>b. A b) = A True \<sqinter> A False" 
44921  375 
by (simp add: UNIV_bool INF_insert inf_commute) 
43868  376 

43873  377 
lemma SUP_UNIV_bool_expand: 
43871  378 
"(\<Squnion>b. A b) = A True \<squnion> A False" 
44921  379 
by (simp add: UNIV_bool SUP_insert sup_commute) 
43871  380 

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381 
end 
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382 

44024  383 
class complete_distrib_lattice = complete_lattice + 
44039  384 
assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" 
44024  385 
assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" 
386 
begin 

387 

44039  388 
lemma sup_INF: 
389 
"a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)" 

390 
by (simp add: INF_def sup_Inf image_image) 

391 

392 
lemma inf_SUP: 

393 
"a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)" 

394 
by (simp add: SUP_def inf_Sup image_image) 

395 

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396 
lemma dual_complete_distrib_lattice: 
44845  397 
"class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
44024  398 
apply (rule class.complete_distrib_lattice.intro) 
399 
apply (fact dual_complete_lattice) 

400 
apply (rule class.complete_distrib_lattice_axioms.intro) 

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401 
apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf) 
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402 
done 
44024  403 

44322  404 
subclass distrib_lattice proof 
44024  405 
fix a b c 
406 
from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" . 

44919  407 
then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def) 
44024  408 
qed 
409 

44039  410 
lemma Inf_sup: 
411 
"\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)" 

412 
by (simp add: sup_Inf sup_commute) 

413 

414 
lemma Sup_inf: 

415 
"\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)" 

416 
by (simp add: inf_Sup inf_commute) 

417 

418 
lemma INF_sup: 

419 
"(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)" 

420 
by (simp add: sup_INF sup_commute) 

421 

422 
lemma SUP_inf: 

423 
"(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)" 

424 
by (simp add: inf_SUP inf_commute) 

425 

426 
lemma Inf_sup_eq_top_iff: 

427 
"(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)" 

428 
by (simp only: Inf_sup INF_top_conv) 

429 

430 
lemma Sup_inf_eq_bot_iff: 

431 
"(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)" 

432 
by (simp only: Sup_inf SUP_bot_conv) 

433 

434 
lemma INF_sup_distrib2: 

435 
"(\<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)" 

436 
by (subst INF_commute) (simp add: sup_INF INF_sup) 

437 

438 
lemma SUP_inf_distrib2: 

439 
"(\<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)" 

440 
by (subst SUP_commute) (simp add: inf_SUP SUP_inf) 

441 

44024  442 
end 
443 

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444 
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice 
43873  445 
begin 
446 

43943  447 
lemma dual_complete_boolean_algebra: 
44845  448 
"class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion>  y) uminus" 
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449 
by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra) 
43943  450 

43873  451 
lemma uminus_Inf: 
452 
" (\<Sqinter>A) = \<Squnion>(uminus ` A)" 

453 
proof (rule antisym) 

454 
show " \<Sqinter>A \<le> \<Squnion>(uminus ` A)" 

455 
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp 

456 
show "\<Squnion>(uminus ` A) \<le>  \<Sqinter>A" 

457 
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto 

458 
qed 

459 

44041  460 
lemma uminus_INF: " (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A.  B x)" 
461 
by (simp add: INF_def SUP_def uminus_Inf image_image) 

462 

43873  463 
lemma uminus_Sup: 
464 
" (\<Squnion>A) = \<Sqinter>(uminus ` A)" 

465 
proof  

466 
have "\<Squnion>A =  \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf) 

467 
then show ?thesis by simp 

468 
qed 

469 

470 
lemma uminus_SUP: " (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A.  B x)" 

471 
by (simp add: INF_def SUP_def uminus_Sup image_image) 

472 

473 
end 

474 

43940  475 
class complete_linorder = linorder + complete_lattice 
476 
begin 

477 

43943  478 
lemma dual_complete_linorder: 
44845  479 
"class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
43943  480 
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) 
481 

44918  482 
lemma Inf_less_iff: 
43940  483 
"\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" 
484 
unfolding not_le [symmetric] le_Inf_iff by auto 

485 

44918  486 
lemma INF_less_iff: 
44041  487 
"(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" 
488 
unfolding INF_def Inf_less_iff by auto 

489 

44918  490 
lemma less_Sup_iff: 
43940  491 
"a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" 
492 
unfolding not_le [symmetric] Sup_le_iff by auto 

493 

44918  494 
lemma less_SUP_iff: 
43940  495 
"a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" 
496 
unfolding SUP_def less_Sup_iff by auto 

497 

44918  498 
lemma Sup_eq_top_iff [simp]: 
43943  499 
"\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" 
500 
proof 

501 
assume *: "\<Squnion>A = \<top>" 

502 
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric] 

503 
proof (intro allI impI) 

504 
fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i" 

505 
unfolding less_Sup_iff by auto 

506 
qed 

507 
next 

508 
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" 

509 
show "\<Squnion>A = \<top>" 

510 
proof (rule ccontr) 

511 
assume "\<Squnion>A \<noteq> \<top>" 

512 
with top_greatest [of "\<Squnion>A"] 

513 
have "\<Squnion>A < \<top>" unfolding le_less by auto 

514 
then have "\<Squnion>A < \<Squnion>A" 

515 
using * unfolding less_Sup_iff by auto 

516 
then show False by auto 

517 
qed 

518 
qed 

519 

44918  520 
lemma SUP_eq_top_iff [simp]: 
44041  521 
"(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)" 
44919  522 
unfolding SUP_def by auto 
44041  523 

44918  524 
lemma Inf_eq_bot_iff [simp]: 
43943  525 
"\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" 
44920  526 
using dual_complete_linorder 
527 
by (rule complete_linorder.Sup_eq_top_iff) 

43943  528 

44918  529 
lemma INF_eq_bot_iff [simp]: 
43967  530 
"(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)" 
44919  531 
unfolding INF_def by auto 
43967  532 

43940  533 
end 
534 

43873  535 

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536 
subsection {* Complete lattice on @{typ bool} *} 
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537 

44024  538 
instantiation bool :: complete_lattice 
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539 
begin 
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540 

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541 
definition 
46154  542 
[simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A" 
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543 

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544 
definition 
46154  545 
[simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A" 
32077
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546 

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547 
instance proof 
44322  548 
qed (auto intro: bool_induct) 
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549 

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550 
end 
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551 

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552 
lemma not_False_in_image_Ball [simp]: 
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553 
"False \<notin> P ` A \<longleftrightarrow> Ball A P" 
a81f95693c68
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haftmann
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changeset

554 
by auto 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
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555 

a81f95693c68
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556 
lemma True_in_image_Bex [simp]: 
a81f95693c68
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haftmann
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557 
"True \<in> P ` A \<longleftrightarrow> Bex A P" 
a81f95693c68
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haftmann
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changeset

558 
by auto 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
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559 

43873  560 
lemma INF_bool_eq [simp]: 
32120
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561 
"INFI = Ball" 
49905
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562 
by (simp add: fun_eq_iff INF_def) 
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563 

43873  564 
lemma SUP_bool_eq [simp]: 
32120
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changeset

565 
"SUPR = Bex" 
49905
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haftmann
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566 
by (simp add: fun_eq_iff SUP_def) 
32120
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567 

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568 
instance bool :: complete_boolean_algebra proof 
44322  569 
qed (auto intro: bool_induct) 
44024  570 

46631
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571 

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572 
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *} 
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573 

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574 
instantiation "fun" :: (type, complete_lattice) complete_lattice 
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575 
begin 
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576 

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577 
definition 
44024  578 
"\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" 
41080  579 

46882  580 
lemma Inf_apply [simp, code]: 
44024  581 
"(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" 
41080  582 
by (simp add: Inf_fun_def) 
32077
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583 

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584 
definition 
44024  585 
"\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" 
41080  586 

46882  587 
lemma Sup_apply [simp, code]: 
44024  588 
"(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" 
41080  589 
by (simp add: Sup_fun_def) 
32077
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590 

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591 
instance proof 
46884  592 
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least) 
32077
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593 

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594 
end 
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595 

46882  596 
lemma INF_apply [simp]: 
41080  597 
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" 
46884  598 
by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def) 
38705  599 

46882  600 
lemma SUP_apply [simp]: 
41080  601 
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" 
46884  602 
by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def) 
32077
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603 

44024  604 
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof 
46884  605 
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image) 
44024  606 

43873  607 
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. 
608 

46631
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609 

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610 
subsection {* Complete lattice on unary and binary predicates *} 
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611 

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612 
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)" 
46884  613 
by simp 
46631
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614 

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615 
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)" 
46884  616 
by simp 
46631
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617 

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618 
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" 
46884  619 
by auto 
46631
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620 

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621 
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" 
46884  622 
by auto 
46631
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623 

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624 
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" 
46884  625 
by auto 
46631
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626 

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627 
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" 
46884  628 
by auto 
46631
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629 

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630 
lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" 
46884  631 
by auto 
46631
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632 

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633 
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" 
46884  634 
by auto 
46631
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635 

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636 
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)" 
46884  637 
by simp 
46631
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638 

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639 
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)" 
46884  640 
by simp 
46631
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641 

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642 
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" 
46884  643 
by auto 
46631
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644 

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645 
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" 
46884  646 
by auto 
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647 

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648 
lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R" 
46884  649 
by auto 
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650 

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651 
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" 
46884  652 
by auto 
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653 

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654 

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655 
subsection {* Complete lattice on @{typ "_ set"} *} 
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656 

45960
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657 
instantiation "set" :: (type) complete_lattice 
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658 
begin 
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659 

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660 
definition 
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661 
"\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}" 
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662 

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663 
definition 
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664 
"\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}" 
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665 

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666 
instance proof 
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667 
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def Inf_bool_def Sup_bool_def le_fun_def) 
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668 

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669 
end 
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670 

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671 
instance "set" :: (type) complete_boolean_algebra 
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672 
proof 
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673 
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def) 
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674 

32077
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675 

46631
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676 
subsubsection {* Inter *} 
41082  677 

678 
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where 

679 
"Inter S \<equiv> \<Sqinter>S" 

680 

681 
notation (xsymbols) 

682 
Inter ("\<Inter>_" [90] 90) 

683 

684 
lemma Inter_eq: 

685 
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" 

686 
proof (rule set_eqI) 

687 
fix x 

688 
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" 

689 
by auto 

690 
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" 

45960
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691 
by (simp add: Inf_set_def image_def) 
41082  692 
qed 
693 

43741  694 
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" 
41082  695 
by (unfold Inter_eq) blast 
696 

43741  697 
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" 
41082  698 
by (simp add: Inter_eq) 
699 

700 
text {* 

701 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

43741  702 
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when 
703 
@{prop "X \<in> C"} does not! This rule is analogous to @{text spec}. 

41082  704 
*} 
705 

43741  706 
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" 
41082  707 
by auto 
708 

43741  709 
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" 
41082  710 
 {* ``Classical'' elimination rule  does not require proving 
43741  711 
@{prop "X \<in> C"}. *} 
41082  712 
by (unfold Inter_eq) blast 
713 

43741  714 
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" 
43740  715 
by (fact Inf_lower) 
716 

41082  717 
lemma Inter_subset: 
43755  718 
"(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" 
43740  719 
by (fact Inf_less_eq) 
41082  720 

43755  721 
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A" 
43740  722 
by (fact Inf_greatest) 
41082  723 

44067  724 
lemma Inter_empty: "\<Inter>{} = UNIV" 
725 
by (fact Inf_empty) (* already simp *) 

41082  726 

44067  727 
lemma Inter_UNIV: "\<Inter>UNIV = {}" 
728 
by (fact Inf_UNIV) (* already simp *) 

41082  729 

44920  730 
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" 
731 
by (fact Inf_insert) (* already simp *) 

41082  732 

733 
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" 

43899  734 
by (fact less_eq_Inf_inter) 
41082  735 

736 
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" 

43756  737 
by (fact Inf_union_distrib) 
738 

43868  739 
lemma Inter_UNIV_conv [simp, no_atp]: 
43741  740 
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" 
741 
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" 

43801  742 
by (fact Inf_top_conv)+ 
41082  743 

43741  744 
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" 
43899  745 
by (fact Inf_superset_mono) 
41082  746 

747 

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748 
subsubsection {* Intersections of families *} 
41082  749 

750 
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 

751 
"INTER \<equiv> INFI" 

752 

43872  753 
text {* 
754 
Note: must use name @{const INTER} here instead of @{text INT} 

755 
to allow the following syntax coexist with the plain constant name. 

756 
*} 

757 

41082  758 
syntax 
759 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

760 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) 

761 

762 
syntax (xsymbols) 

763 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

764 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) 

765 

766 
syntax (latex output) 

767 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

768 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) 

769 

770 
translations 

771 
"INT x y. B" == "INT x. INT y. B" 

772 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

773 
"INT x. B" == "INT x:CONST UNIV. B" 

774 
"INT x:A. B" == "CONST INTER A (%x. B)" 

775 

776 
print_translation {* 

42284  777 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] 
41082  778 
*}  {* to avoid etacontraction of body *} 
779 

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780 
lemma INTER_eq: 
41082  781 
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" 
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782 
by (auto simp add: INF_def) 
41082  783 

784 
lemma Inter_image_eq [simp]: 

785 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 

43872  786 
by (rule sym) (fact INF_def) 
41082  787 

43817  788 
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

789 
by (auto simp add: INF_def image_def) 
41082  790 

43817  791 
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

792 
by (auto simp add: INF_def image_def) 
41082  793 

43852  794 
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" 
41082  795 
by auto 
796 

43852  797 
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" 
798 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a\<in>A"}. *} 

44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

799 
by (auto simp add: INF_def image_def) 
41082  800 

801 
lemma INT_cong [cong]: 

43854  802 
"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)" 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

803 
by (fact INF_cong) 
41082  804 

805 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 

806 
by blast 

807 

808 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 

809 
by blast 

810 

43817  811 
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

812 
by (fact INF_lower) 
41082  813 

43817  814 
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

815 
by (fact INF_greatest) 
41082  816 

44067  817 
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

818 
by (fact INF_empty) 
43854  819 

43817  820 
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" 
43872  821 
by (fact INF_absorb) 
41082  822 

43854  823 
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" 
41082  824 
by (fact le_INF_iff) 
825 

826 
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" 

43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

827 
by (fact INF_insert) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

828 

db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

829 
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

830 
by (fact INF_union) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

831 

db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

832 
lemma INT_insert_distrib: 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

833 
"u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

834 
by blast 
43854  835 

41082  836 
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

837 
by (fact INF_constant) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

838 

44920  839 
lemma INTER_UNIV_conv: 
43817  840 
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" 
841 
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" 

44920  842 
by (fact INF_top_conv)+ (* already simp *) 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

843 

db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

844 
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" 
43873  845 
by (fact INF_UNIV_bool_expand) 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

846 

db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

847 
lemma INT_anti_mono: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

848 
"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)" 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

849 
 {* The last inclusion is POSITIVE! *} 
43940  850 
by (fact INF_superset_mono) 
41082  851 

852 
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" 

853 
by blast 

854 

43817  855 
lemma vimage_INT: "f ` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f ` B x)" 
41082  856 
by blast 
857 

858 

46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset

859 
subsubsection {* Union *} 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

860 

32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

861 
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where 
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

862 
"Union S \<equiv> \<Squnion>S" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

863 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

864 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

865 
Union ("\<Union>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

866 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

867 
lemma Union_eq: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

868 
"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
38705
diff
changeset

869 
proof (rule set_eqI) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

870 
fix x 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

871 
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

872 
by auto 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

873 
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" 
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

874 
by (simp add: Sup_set_def image_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

875 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

876 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

877 
lemma Union_iff [simp, no_atp]: 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

878 
"A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

879 
by (unfold Union_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

880 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

881 
lemma UnionI [intro]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

882 
"X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

883 
 {* The order of the premises presupposes that @{term C} is rigid; 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

884 
@{term A} may be flexible. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

885 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

886 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

887 
lemma UnionE [elim!]: 
43817  888 
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

889 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

890 

43817  891 
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" 
43901  892 
by (fact Sup_upper) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

893 

43817  894 
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" 
43901  895 
by (fact Sup_least) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

896 

44920  897 
lemma Union_empty: "\<Union>{} = {}" 
898 
by (fact Sup_empty) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

899 

44920  900 
lemma Union_UNIV: "\<Union>UNIV = UNIV" 
901 
by (fact Sup_UNIV) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

902 

44920  903 
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B" 
904 
by (fact Sup_insert) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

905 

43817  906 
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" 
43901  907 
by (fact Sup_union_distrib) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

908 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

909 
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" 
43901  910 
by (fact Sup_inter_less_eq) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

911 

44920  912 
lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" 
913 
by (fact Sup_bot_conv) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

914 

44920  915 
lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" 
916 
by (fact Sup_bot_conv) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

917 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

918 
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

919 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

920 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

921 
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

922 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

923 

43817  924 
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" 
43901  925 
by (fact Sup_subset_mono) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

926 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

927 

46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset

928 
subsubsection {* Unions of families *} 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

929 

32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

930 
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

931 
"UNION \<equiv> SUPR" 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

932 

43872  933 
text {* 
934 
Note: must use name @{const UNION} here instead of @{text UN} 

935 
to allow the following syntax coexist with the plain constant name. 

936 
*} 

937 

32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

938 
syntax 
35115  939 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset

940 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10) 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

941 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

942 
syntax (xsymbols) 
35115  943 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset

944 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

945 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

946 
syntax (latex output) 
35115  947 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset

948 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

949 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

950 
translations 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

951 
"UN x y. B" == "UN x. UN y. B" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

952 
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

953 
"UN x. B" == "UN x:CONST UNIV. B" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

954 
"UN x:A. B" == "CONST UNION A (%x. B)" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

955 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

956 
text {* 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

957 
Note the difference between ordinary xsymbol syntax of indexed 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

958 
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

959 
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

960 
former does not make the index expression a subscript of the 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

961 
union/intersection symbol because this leads to problems with nested 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

962 
subscripts in Proof General. 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

963 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

964 

35115  965 
print_translation {* 
42284  966 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] 
35115  967 
*}  {* to avoid etacontraction of body *} 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

968 

44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

969 
lemma UNION_eq [no_atp]: 
32135
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set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

970 
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

971 
by (auto simp add: SUP_def) 
44920  972 

45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

973 
lemma bind_UNION [code]: 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

974 
"Set.bind A f = UNION A f" 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

975 
by (simp add: bind_def UNION_eq) 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

976 

46036  977 
lemma member_bind [simp]: 
978 
"x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f " 

979 
by (simp add: bind_UNION) 

980 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

981 
lemma Union_image_eq [simp]: 
43817  982 
"\<Union>(B ` A) = (\<Union>x\<in>A. B x)" 
44920  983 
by (rule sym) (fact SUP_def) 
984 

46036  985 
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

986 
by (auto simp add: SUP_def image_def) 
11979  987 

43852  988 
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" 
11979  989 
 {* The order of the premises presupposes that @{term A} is rigid; 
990 
@{term b} may be flexible. *} 

991 
by auto 

992 

43852  993 
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" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

994 
by (auto simp add: SUP_def image_def) 
923  995 

11979  996 
lemma UN_cong [cong]: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

997 
"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)" 
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

998 
by (fact SUP_cong) 
11979  999 

29691  1000 
lemma strong_UN_cong: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1001 
"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)" 
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1002 
by (unfold simp_implies_def) (fact UN_cong) 
29691  1003 

43817  1004 
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})" 
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

1005 
by blast 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

1006 

43817  1007 
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

1008 
by (fact SUP_upper) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1009 

43817  1010 
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

1011 
by (fact SUP_least) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1012 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

1013 
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1014 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1015 

43817  1016 
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1017 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1018 

44067  1019 
lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}" 
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

1020 
by (fact SUP_empty) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1021 

44920  1022 
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" 
1023 
by (fact SUP_bot) (* already simp *) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1024 

43817  1025 
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1026 
by (fact SUP_absorb) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1027 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1028 
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1029 
by (fact SUP_insert) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1030 

44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset

1031 
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)" 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1032 
by (fact SUP_union) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1033 

43967  1034 
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)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1035 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1036 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1037 
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" 
35629  1038 
by (fact SUP_le_iff) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1039 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1040 
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1041 
by (fact SUP_constant) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1042 

43944  1043 
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1044 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1045 

44920  1046 
lemma UNION_empty_conv: 
43817  1047 
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" 
1048 
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" 

44920  1049 
by (fact SUP_bot_conv)+ (* already simp *) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1050 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

1051 
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1052 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1053 

43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1054 
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1055 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1056 

43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1057 
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1058 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1059 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1060 
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1061 
by (auto simp add: split_if_mem2) 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1062 

43817  1063 
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)" 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1064 
by (fact SUP_UNIV_bool_expand) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1065 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1066 
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1067 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1068 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1069 
lemma UN_mono: 
43817  1070 
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1071 
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" 
43940  1072 
by (fact SUP_subset_mono) 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1073 

43817  1074 
lemma vimage_Union: "f ` (\<Union>A) = (\<Union>X\<in>A. f ` X)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1075 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1076 

43817  1077 
lemma vimage_UN: "f ` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f ` B x)" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1078 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1079 

43817  1080 
lemma vimage_eq_UN: "f ` B = (\<Union>y\<in>B. f ` {y})" 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1081 
 {* NOT suitable for rewriting *} 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1082 
by blast 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1083 

43817  1084 
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" 
1085 
by blast 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1086 

45013  1087 
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" 
1088 
by blast 

1089 

11979  1090 

46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset

1091 
subsubsection {* Distributive laws *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1092 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1093 
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1094 
by (fact inf_Sup) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1095 

44039  1096 
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" 
1097 
by (fact sup_Inf) 

1098 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1099 
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" 
44039  1100 
by (fact Sup_inf) 
1101 

1102 
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)" 

1103 
by (rule sym) (rule INF_inf_distrib) 

1104 

1105 
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)" 

1106 
by (rule sym) (rule SUP_sup_distrib) 

1107 

1108 
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" 

1109 
by (simp only: INT_Int_distrib INF_def) 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1110 

43817  1111 
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1112 
 {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1113 
 {* Union of a family of unions *} 
44039  1114 
by (simp only: UN_Un_distrib SUP_def) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1115 

44039  1116 
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" 
1117 
by (fact sup_INF) 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1118 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1119 
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1120 
 {* Halmos, Naive Set Theory, page 35. *} 
44039  1121 
by (fact inf_SUP) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1122 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1123 
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)" 
44039  1124 
by (fact SUP_inf_distrib2) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1125 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1126 
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)" 
44039  1127 
by (fact INF_sup_distrib2) 
1128 

1129 
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" 

1130 
by (fact Sup_inf_eq_bot_iff) 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1131 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1132 

46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset

1133 
subsubsection {* Complement *} 
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1134 

43873  1135 
lemma Compl_INT [simp]: " (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. B x)" 
1136 
by (fact uminus_INF) 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1137 

43873  1138 
lemma Compl_UN [simp]: " (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. B x)" 
1139 
by (fact uminus_SUP) 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1140 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1141 

46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset

1142 
subsubsection {* Miniscoping and maxiscoping *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1143 

13860  1144 
text {* \medskip Miniscoping: pushing in quantifiers and big Unions 
1145 
and Intersections. *} 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1146 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1147 
lemma UN_simps [simp]: 
43817  1148 
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1149 
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))" 
43852  1150 
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))" 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1151 
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" 
43852  1152 
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" 
1153 
"\<And>A B C. (\<Union>x\<in>C. A x  B) = ((\<Union>x\<in>C. A x)  B)" 

1154 
"\<And>A B C. (\<Union>x\<in>C. A  B x) = (A  (\<Inter>x\<in>C. B x))" 

1155 
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" 

1156 
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" 

43831  1157 
"\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1158 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1159 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1160 
lemma INT_simps [simp]: 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1161 
"\<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)" 
43831  1162 
"\<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))" 
43852  1163 
"\<And>A B C. (\<Inter>x\<in>C. A x  B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x)  B)" 
1164 
"\<And>A B C. (\<Inter>x\<in>C. A  B x) = (if C={} then UNIV else A  (\<Union>x\<in>C. B x))" 

43817  1165 
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" 
43852  1166 
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" 
1167 
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" 

1168 
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" 

1169 
"\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" 

1170 
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))" 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1171 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1172 

43967  1173 
lemma UN_ball_bex_simps [simp, no_atp]: 
43852  1174 
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" 
43967  1175 
"\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" 
43852  1176 
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" 
1177 
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1178 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1179 

43943  1180 

13860  1181 
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} 
1182 

1183 
lemma UN_extend_simps: 

43817  1184 
"\<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)))" 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1185 
"\<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))" 
43852  1186 
"\<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))" 
1187 
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" 

1188 
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" 

43817  1189 
"\<And>A B C. ((\<Union>x\<in>C. A x)  B) = (\<Union>x\<in>C. A x  B)" 
1190 
"\<And>A B C. (A  (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A  B x)" 

43852  1191 
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" 
1192 
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)" 

43831  1193 
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" 
13860  1194 
by auto 
1195 

1196 
lemma INT_extend_simps: 

43852  1197 
"\<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))" 
1198 
"\<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))" 

1199 
"\<And>A B C. (\<Inter>x\<in>C. A x)  B = (if C={} then UNIV  B else (\<Inter>x\<in>C. A x  B))" 

1200 
"\<And>A B C. A  (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A  B x))" 

43817  1201 
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" 
43852  1202 
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" 
1203 
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" 

1204 
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" 

1205 
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)" 

1206 
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" 

13860  1207 
by auto 
1208 

43872  1209 
text {* Finally *} 
1210 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1211 
no_notation 
46691  1212 
less_eq (infix "\<sqsubseteq>" 50) and 
1213 
less (infix "\<sqsubset>" 50) 

32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

1214 

30596  1215 
lemmas mem_simps = 
1216 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 
