author  hoelzl 
Wed, 20 Feb 2013 12:04:42 +0100  
changeset 51328  d63ec23c9125 
parent 49905  a81f95693c68 
child 51341  8c10293e7ea7 
permissions  rwrr 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

1 
(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) 
11979  2 

44104  3 
header {* Complete lattices *} 
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

4 

44860
56101fa00193
renamed theory Complete_Lattice to Complete_Lattices, in accordance with Lattices, Orderings etc.
haftmann
parents:
44845
diff
changeset

5 
theory Complete_Lattices 
32139  6 
imports Set 
7 
begin 

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

8 

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

9 
notation 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

10 
less_eq (infix "\<sqsubseteq>" 50) and 
46691  11 
less (infix "\<sqsubset>" 50) 
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

12 

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) 

21 

46691  22 

32139  23 
subsection {* Abstract complete lattices *} 
24 

34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

25 
class complete_lattice = bounded_lattice + Inf + Sup + 
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

26 
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

27 
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

28 
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

29 
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

30 
begin 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

31 

32678  32 
lemma dual_complete_lattice: 
44845  33 
"class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
36364
diff
changeset

34 
by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

35 
(unfold_locales, (fact bot_least top_greatest 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

36 
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)" 

43 

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 

51 

52 
syntax 

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

54 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) 

55 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) 

56 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) 

57 

58 
syntax (xsymbols) 

59 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 

60 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

61 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 

62 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

63 

64 
translations 

65 
"INF x y. B" == "INF x. INF y. B" 

66 
"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 

67 
"INF x. B" == "INF x:CONST UNIV. B" 

68 
"INF x:A. B" == "CONST INFI A (%x. B)" 

69 
"SUP x y. B" == "SUP x. SUP y. B" 

70 
"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 

71 
"SUP x. B" == "SUP x:CONST UNIV. B" 

72 
"SUP x:A. B" == "CONST SUPR A (%x. B)" 

73 

74 
print_translation {* 

75 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, 

76 
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] 

77 
*}  {* to avoid etacontraction of body *} 

78 

79 
context complete_lattice 

80 
begin 

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

81 

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 
90 
complete_lattice.INF_def [OF dual_complete_lattice]) 

44040  91 

51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

92 
lemma Sup_eqI: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

93 
"(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

94 
by (blast intro: antisym Sup_least Sup_upper) 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

95 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

96 
lemma Inf_eqI: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

97 
"(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

98 
by (blast intro: antisym Inf_greatest Inf_lower) 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

99 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

100 
lemma SUP_eqI: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

101 
"(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

102 
unfolding SUP_def by (rule Sup_eqI) auto 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

103 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

104 
lemma INF_eqI: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

105 
"(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

106 
unfolding INF_def by (rule Inf_eqI) auto 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

107 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

108 
lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i" 
44040  109 
by (auto simp add: INF_def intro: Inf_lower) 
110 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

111 
lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)" 
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

112 
by (auto simp add: INF_def intro: Inf_greatest) 
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

113 

cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

114 
lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)" 
44040  115 
by (auto simp add: SUP_def intro: Sup_upper) 
116 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

117 
lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u" 
44040  118 
by (auto simp add: SUP_def intro: Sup_least) 
119 

120 
lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" 

121 
using Inf_lower [of u A] by auto 

122 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

123 
lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u" 
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

124 
using INF_lower [of i A f] by auto 
44040  125 

126 
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A" 

127 
using Sup_upper [of u A] by auto 

128 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

129 
lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)" 
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

130 
using SUP_upper [of i A f] by auto 
44040  131 

44918  132 
lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" 
44040  133 
by (auto intro: Inf_greatest dest: Inf_lower) 
134 

44918  135 
lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)" 
44040  136 
by (auto simp add: INF_def le_Inf_iff) 
137 

44918  138 
lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" 
44040  139 
by (auto intro: Sup_least dest: Sup_upper) 
140 

44918  141 
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)" 
44040  142 
by (auto simp add: SUP_def Sup_le_iff) 
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

143 

41080  144 
lemma Inf_empty [simp]: 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

145 
"\<Sqinter>{} = \<top>" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

146 
by (auto intro: antisym Inf_greatest) 
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

147 

44067  148 
lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" 
44040  149 
by (simp add: INF_def) 
150 

41080  151 
lemma Sup_empty [simp]: 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

152 
"\<Squnion>{} = \<bottom>" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

153 
by (auto intro: antisym Sup_least) 
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

154 

44067  155 
lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" 
44040  156 
by (simp add: SUP_def) 
157 

41080  158 
lemma Inf_UNIV [simp]: 
159 
"\<Sqinter>UNIV = \<bottom>" 

44040  160 
by (auto intro!: antisym Inf_lower) 
41080  161 

162 
lemma Sup_UNIV [simp]: 

163 
"\<Squnion>UNIV = \<top>" 

44040  164 
by (auto intro!: antisym Sup_upper) 
41080  165 

44918  166 
lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 
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

167 
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

168 

44040  169 
lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f" 
44919  170 
by (simp add: INF_def) 
44040  171 

44918  172 
lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 
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

173 
by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

174 

44040  175 
lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f" 
44919  176 
by (simp add: SUP_def) 
44040  177 

44918  178 
lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))" 
44068  179 
by (simp add: INF_def image_image) 
180 

44918  181 
lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))" 
44068  182 
by (simp add: SUP_def image_image) 
183 

44040  184 
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" 
185 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

186 

187 
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" 

188 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 

189 

43899  190 
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" 
191 
by (auto intro: Inf_greatest Inf_lower) 

192 

193 
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B" 

194 
by (auto intro: Sup_least Sup_upper) 

195 

44041  196 
lemma INF_cong: 
197 
"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)" 

198 
by (simp add: INF_def image_def) 

199 

200 
lemma SUP_cong: 

201 
"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)" 

202 
by (simp add: SUP_def image_def) 

203 

38705  204 
lemma Inf_mono: 
41971  205 
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" 
43741  206 
shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B" 
38705  207 
proof (rule Inf_greatest) 
208 
fix b assume "b \<in> B" 

41971  209 
with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast 
43741  210 
from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) 
211 
with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto 

38705  212 
qed 
213 

44041  214 
lemma INF_mono: 
215 
"(\<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  216 
unfolding INF_def by (rule Inf_mono) fast 
44041  217 

41082  218 
lemma Sup_mono: 
41971  219 
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" 
43741  220 
shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" 
41082  221 
proof (rule Sup_least) 
222 
fix a assume "a \<in> A" 

41971  223 
with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast 
43741  224 
from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) 
225 
with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto 

41082  226 
qed 
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

227 

44041  228 
lemma SUP_mono: 
229 
"(\<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  230 
unfolding SUP_def by (rule Sup_mono) fast 
44041  231 

232 
lemma INF_superset_mono: 

233 
"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)" 

234 
 {* The last inclusion is POSITIVE! *} 

235 
by (blast intro: INF_mono dest: subsetD) 

236 

237 
lemma SUP_subset_mono: 

238 
"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)" 

239 
by (blast intro: SUP_mono dest: subsetD) 

240 

43868  241 
lemma Inf_less_eq: 
242 
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" 

243 
and "A \<noteq> {}" 

244 
shows "\<Sqinter>A \<sqsubseteq> u" 

245 
proof  

246 
from `A \<noteq> {}` obtain v where "v \<in> A" by blast 

247 
moreover with assms have "v \<sqsubseteq> u" by blast 

248 
ultimately show ?thesis by (rule Inf_lower2) 

249 
qed 

250 

251 
lemma less_eq_Sup: 

252 
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v" 

253 
and "A \<noteq> {}" 

254 
shows "u \<sqsubseteq> \<Squnion>A" 

255 
proof  

256 
from `A \<noteq> {}` obtain v where "v \<in> A" by blast 

257 
moreover with assms have "u \<sqsubseteq> v" by blast 

258 
ultimately show ?thesis by (rule Sup_upper2) 

259 
qed 

260 

51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

261 
lemma SUPR_eq: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

262 
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

263 
assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

264 
shows "(SUP i:A. f i) = (SUP j:B. g j)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

265 
by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+ 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

266 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

267 
lemma INFI_eq: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

268 
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

269 
assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

270 
shows "(INF i:A. f i) = (INF j:B. g j)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

271 
by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+ 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

272 

43899  273 
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" 
43868  274 
by (auto intro: Inf_greatest Inf_lower) 
275 

43899  276 
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B " 
43868  277 
by (auto intro: Sup_least Sup_upper) 
278 

279 
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" 

280 
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) 

281 

44041  282 
lemma INF_union: 
283 
"(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

284 
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) 
44041  285 

43868  286 
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" 
287 
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) 

288 

44041  289 
lemma SUP_union: 
290 
"(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

291 
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) 
44041  292 

293 
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)" 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

294 
by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) 
44041  295 

44918  296 
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") 
297 
proof (rule antisym) 

298 
show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) 

299 
next 

300 
show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) 

301 
qed 

44041  302 

44918  303 
lemma Inf_top_conv [simp, no_atp]: 
43868  304 
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 
305 
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 

306 
proof  

307 
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" 

308 
proof 

309 
assume "\<forall>x\<in>A. x = \<top>" 

310 
then have "A = {} \<or> A = {\<top>}" by auto 

44919  311 
then show "\<Sqinter>A = \<top>" by auto 
43868  312 
next 
313 
assume "\<Sqinter>A = \<top>" 

314 
show "\<forall>x\<in>A. x = \<top>" 

315 
proof (rule ccontr) 

316 
assume "\<not> (\<forall>x\<in>A. x = \<top>)" 

317 
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast 

318 
then obtain B where "A = insert x B" by blast 

44919  319 
with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp 
43868  320 
qed 
321 
qed 

322 
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto 

323 
qed 

324 

44918  325 
lemma INF_top_conv [simp]: 
44041  326 
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" 
327 
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" 

44919  328 
by (auto simp add: INF_def) 
44041  329 

44918  330 
lemma Sup_bot_conv [simp, no_atp]: 
43868  331 
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P) 
332 
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q) 

44920  333 
using dual_complete_lattice 
334 
by (rule complete_lattice.Inf_top_conv)+ 

43868  335 

44918  336 
lemma SUP_bot_conv [simp]: 
44041  337 
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" 
338 
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" 

44919  339 
by (auto simp add: SUP_def) 
44041  340 

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

341 
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

342 
by (auto intro: antisym INF_lower INF_greatest) 
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

343 

43870  344 
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

345 
by (auto intro: antisym SUP_upper SUP_least) 
43870  346 

44918  347 
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" 
44921  348 
by (cases "A = {}") simp_all 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

349 

44918  350 
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" 
44921  351 
by (cases "A = {}") simp_all 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

352 

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

353 
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)" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

354 
by (iprover intro: INF_lower INF_greatest order_trans antisym) 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

355 

43870  356 
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)" 
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

357 
by (iprover intro: SUP_upper SUP_least order_trans antisym) 
43870  358 

43871  359 
lemma INF_absorb: 
43868  360 
assumes "k \<in> I" 
361 
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" 

362 
proof  

363 
from assms obtain J where "I = insert k J" by blast 

364 
then show ?thesis by (simp add: INF_insert) 

365 
qed 

366 

43871  367 
lemma SUP_absorb: 
368 
assumes "k \<in> I" 

369 
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" 

370 
proof  

371 
from assms obtain J where "I = insert k J" by blast 

372 
then show ?thesis by (simp add: SUP_insert) 

373 
qed 

374 

375 
lemma INF_constant: 

43868  376 
"(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" 
44921  377 
by simp 
43868  378 

43871  379 
lemma SUP_constant: 
380 
"(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" 

44921  381 
by simp 
43871  382 

43943  383 
lemma less_INF_D: 
384 
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i" 

385 
proof  

386 
note `y < (\<Sqinter>i\<in>A. f i)` 

387 
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A` 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

388 
by (rule INF_lower) 
43943  389 
finally show "y < f i" . 
390 
qed 

391 

392 
lemma SUP_lessD: 

393 
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y" 

394 
proof  

395 
have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A` 

44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset

396 
by (rule SUP_upper) 
43943  397 
also note `(\<Squnion>i\<in>A. f i) < y` 
398 
finally show "f i < y" . 

399 
qed 

400 

43873  401 
lemma INF_UNIV_bool_expand: 
43868  402 
"(\<Sqinter>b. A b) = A True \<sqinter> A False" 
44921  403 
by (simp add: UNIV_bool INF_insert inf_commute) 
43868  404 

43873  405 
lemma SUP_UNIV_bool_expand: 
43871  406 
"(\<Squnion>b. A b) = A True \<squnion> A False" 
44921  407 
by (simp add: UNIV_bool SUP_insert sup_commute) 
43871  408 

51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

409 
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

410 
by (blast intro: Sup_upper2 Inf_lower ex_in_conv) 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

411 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

412 
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

413 
unfolding INF_def SUP_def by (rule Inf_le_Sup) auto 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

414 

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

415 
end 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

416 

44024  417 
class complete_distrib_lattice = complete_lattice + 
44039  418 
assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" 
44024  419 
assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" 
420 
begin 

421 

44039  422 
lemma sup_INF: 
423 
"a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)" 

424 
by (simp add: INF_def sup_Inf image_image) 

425 

426 
lemma inf_SUP: 

427 
"a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)" 

428 
by (simp add: SUP_def inf_Sup image_image) 

429 

44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

430 
lemma dual_complete_distrib_lattice: 
44845  431 
"class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
44024  432 
apply (rule class.complete_distrib_lattice.intro) 
433 
apply (fact dual_complete_lattice) 

434 
apply (rule class.complete_distrib_lattice_axioms.intro) 

44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

435 
apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf) 
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

436 
done 
44024  437 

44322  438 
subclass distrib_lattice proof 
44024  439 
fix a b c 
440 
from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" . 

44919  441 
then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def) 
44024  442 
qed 
443 

44039  444 
lemma Inf_sup: 
445 
"\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)" 

446 
by (simp add: sup_Inf sup_commute) 

447 

448 
lemma Sup_inf: 

449 
"\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)" 

450 
by (simp add: inf_Sup inf_commute) 

451 

452 
lemma INF_sup: 

453 
"(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)" 

454 
by (simp add: sup_INF sup_commute) 

455 

456 
lemma SUP_inf: 

457 
"(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)" 

458 
by (simp add: inf_SUP inf_commute) 

459 

460 
lemma Inf_sup_eq_top_iff: 

461 
"(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)" 

462 
by (simp only: Inf_sup INF_top_conv) 

463 

464 
lemma Sup_inf_eq_bot_iff: 

465 
"(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)" 

466 
by (simp only: Sup_inf SUP_bot_conv) 

467 

468 
lemma INF_sup_distrib2: 

469 
"(\<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)" 

470 
by (subst INF_commute) (simp add: sup_INF INF_sup) 

471 

472 
lemma SUP_inf_distrib2: 

473 
"(\<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)" 

474 
by (subst SUP_commute) (simp add: inf_SUP SUP_inf) 

475 

44024  476 
end 
477 

44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

478 
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice 
43873  479 
begin 
480 

43943  481 
lemma dual_complete_boolean_algebra: 
44845  482 
"class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion>  y) uminus" 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

483 
by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra) 
43943  484 

43873  485 
lemma uminus_Inf: 
486 
" (\<Sqinter>A) = \<Squnion>(uminus ` A)" 

487 
proof (rule antisym) 

488 
show " \<Sqinter>A \<le> \<Squnion>(uminus ` A)" 

489 
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp 

490 
show "\<Squnion>(uminus ` A) \<le>  \<Sqinter>A" 

491 
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto 

492 
qed 

493 

44041  494 
lemma uminus_INF: " (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A.  B x)" 
495 
by (simp add: INF_def SUP_def uminus_Inf image_image) 

496 

43873  497 
lemma uminus_Sup: 
498 
" (\<Squnion>A) = \<Sqinter>(uminus ` A)" 

499 
proof  

500 
have "\<Squnion>A =  \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf) 

501 
then show ?thesis by simp 

502 
qed 

503 

504 
lemma uminus_SUP: " (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A.  B x)" 

505 
by (simp add: INF_def SUP_def uminus_Sup image_image) 

506 

507 
end 

508 

43940  509 
class complete_linorder = linorder + complete_lattice 
510 
begin 

511 

43943  512 
lemma dual_complete_linorder: 
44845  513 
"class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" 
43943  514 
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) 
515 

44918  516 
lemma Inf_less_iff: 
43940  517 
"\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" 
518 
unfolding not_le [symmetric] le_Inf_iff by auto 

519 

44918  520 
lemma INF_less_iff: 
44041  521 
"(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" 
522 
unfolding INF_def Inf_less_iff by auto 

523 

44918  524 
lemma less_Sup_iff: 
43940  525 
"a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" 
526 
unfolding not_le [symmetric] Sup_le_iff by auto 

527 

44918  528 
lemma less_SUP_iff: 
43940  529 
"a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" 
530 
unfolding SUP_def less_Sup_iff by auto 

531 

44918  532 
lemma Sup_eq_top_iff [simp]: 
43943  533 
"\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" 
534 
proof 

535 
assume *: "\<Squnion>A = \<top>" 

536 
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric] 

537 
proof (intro allI impI) 

538 
fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i" 

539 
unfolding less_Sup_iff by auto 

540 
qed 

541 
next 

542 
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" 

543 
show "\<Squnion>A = \<top>" 

544 
proof (rule ccontr) 

545 
assume "\<Squnion>A \<noteq> \<top>" 

546 
with top_greatest [of "\<Squnion>A"] 

547 
have "\<Squnion>A < \<top>" unfolding le_less by auto 

548 
then have "\<Squnion>A < \<Squnion>A" 

549 
using * unfolding less_Sup_iff by auto 

550 
then show False by auto 

551 
qed 

552 
qed 

553 

44918  554 
lemma SUP_eq_top_iff [simp]: 
44041  555 
"(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)" 
44919  556 
unfolding SUP_def by auto 
44041  557 

44918  558 
lemma Inf_eq_bot_iff [simp]: 
43943  559 
"\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" 
44920  560 
using dual_complete_linorder 
561 
by (rule complete_linorder.Sup_eq_top_iff) 

43943  562 

44918  563 
lemma INF_eq_bot_iff [simp]: 
43967  564 
"(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)" 
44919  565 
unfolding INF_def by auto 
43967  566 

51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

567 
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

568 
proof safe 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

569 
fix y assume "x \<le> \<Squnion>A" "y < x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

570 
then have "y < \<Squnion>A" by auto 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

571 
then show "\<exists>a\<in>A. y < a" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

572 
unfolding less_Sup_iff . 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

573 
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper) 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

574 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

575 
lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

576 
unfolding le_Sup_iff SUP_def by simp 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

577 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

578 
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

579 
proof safe 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

580 
fix y assume "x \<ge> \<Sqinter>A" "y > x" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

581 
then have "y > \<Sqinter>A" by auto 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

582 
then show "\<exists>a\<in>A. y > a" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

583 
unfolding Inf_less_iff . 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

584 
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower) 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

585 

d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

586 
lemma INF_le_iff: 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

587 
"INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)" 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

588 
unfolding Inf_le_iff INF_def by simp 
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset

589 

43940  590 
end 
591 

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

592 
subsection {* Complete lattice on @{typ bool} *} 
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

593 

44024  594 
instantiation bool :: complete_lattice 
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

595 
begin 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

596 

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

597 
definition 
46154  598 
[simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A" 
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

599 

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

600 
definition 
46154  601 
[simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A" 
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

602 

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

603 
instance proof 
44322  604 
qed (auto intro: bool_induct) 
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

605 

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

606 
end 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

607 

49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

608 
lemma not_False_in_image_Ball [simp]: 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

609 
"False \<notin> P ` A \<longleftrightarrow> Ball A P" 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

610 
by auto 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

611 

a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

612 
lemma True_in_image_Bex [simp]: 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

613 
"True \<in> P ` A \<longleftrightarrow> Bex A P" 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

614 
by auto 
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

615 

43873  616 
lemma INF_bool_eq [simp]: 
32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

617 
"INFI = Ball" 
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

618 
by (simp add: fun_eq_iff INF_def) 
32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

619 

43873  620 
lemma SUP_bool_eq [simp]: 
32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

621 
"SUPR = Bex" 
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset

622 
by (simp add: fun_eq_iff SUP_def) 
32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

623 

44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

624 
instance bool :: complete_boolean_algebra proof 
44322  625 
qed (auto intro: bool_induct) 
44024  626 

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

627 

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

628 
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *} 
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

629 

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

630 
instantiation "fun" :: (type, complete_lattice) complete_lattice 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

631 
begin 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

632 

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

633 
definition 
44024  634 
"\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" 
41080  635 

46882  636 
lemma Inf_apply [simp, code]: 
44024  637 
"(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" 
41080  638 
by (simp add: Inf_fun_def) 
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

639 

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

640 
definition 
44024  641 
"\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" 
41080  642 

46882  643 
lemma Sup_apply [simp, code]: 
44024  644 
"(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" 
41080  645 
by (simp add: Sup_fun_def) 
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

646 

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

647 
instance proof 
46884  648 
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least) 
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

649 

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

650 
end 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

651 

46882  652 
lemma INF_apply [simp]: 
41080  653 
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" 
46884  654 
by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def) 
38705  655 

46882  656 
lemma SUP_apply [simp]: 
41080  657 
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" 
46884  658 
by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def) 
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

659 

44024  660 
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof 
46884  661 
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image) 
44024  662 

43873  663 
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. 
664 

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

665 

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

666 
subsection {* Complete lattice on unary and binary predicates *} 
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

667 

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

668 
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)" 
46884  669 
by simp 
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

670 

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

671 
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)" 
46884  672 
by simp 
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

673 

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

674 
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" 
46884  675 
by auto 
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

676 

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

677 
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" 
46884  678 
by auto 
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

679 

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

680 
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" 
46884  681 
by auto 
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

682 

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

683 
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" 
46884  684 
by auto 
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

685 

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

686 
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  687 
by auto 
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

688 

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

689 
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  690 
by auto 
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

691 

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

692 
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)" 
46884  693 
by simp 
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

694 

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

695 
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)" 
46884  696 
by simp 
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

697 

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

698 
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" 
46884  699 
by auto 
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

700 

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

701 
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" 
46884  702 
by auto 
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

703 

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

704 
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  705 
by auto 
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

706 

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

707 
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  708 
by auto 
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

709 

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

710 

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

711 
subsection {* Complete lattice on @{typ "_ set"} *} 
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

712 

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

713 
instantiation "set" :: (type) complete_lattice 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

714 
begin 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

715 

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

716 
definition 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

717 
"\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}" 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

718 

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

719 
definition 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

720 
"\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}" 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

721 

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

722 
instance proof 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

723 
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def Inf_bool_def Sup_bool_def le_fun_def) 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

724 

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

725 
end 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

726 

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

727 
instance "set" :: (type) complete_boolean_algebra 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

728 
proof 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

729 
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def) 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

730 

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

731 

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

732 
subsubsection {* Inter *} 
41082  733 

734 
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where 

735 
"Inter S \<equiv> \<Sqinter>S" 

736 

737 
notation (xsymbols) 

738 
Inter ("\<Inter>_" [90] 90) 

739 

740 
lemma Inter_eq: 

741 
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" 

742 
proof (rule set_eqI) 

743 
fix x 

744 
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" 

745 
by auto 

746 
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>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

747 
by (simp add: Inf_set_def image_def) 
41082  748 
qed 
749 

43741  750 
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" 
41082  751 
by (unfold Inter_eq) blast 
752 

43741  753 
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" 
41082  754 
by (simp add: Inter_eq) 
755 

756 
text {* 

757 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

43741  758 
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when 
759 
@{prop "X \<in> C"} does not! This rule is analogous to @{text spec}. 

41082  760 
*} 
761 

43741  762 
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" 
41082  763 
by auto 
764 

43741  765 
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" 
41082  766 
 {* ``Classical'' elimination rule  does not require proving 
43741  767 
@{prop "X \<in> C"}. *} 
41082  768 
by (unfold Inter_eq) blast 
769 

43741  770 
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" 
43740  771 
by (fact Inf_lower) 
772 

41082  773 
lemma Inter_subset: 
43755  774 
"(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" 
43740  775 
by (fact Inf_less_eq) 
41082  776 

43755  777 
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A" 
43740  778 
by (fact Inf_greatest) 
41082  779 

44067  780 
lemma Inter_empty: "\<Inter>{} = UNIV" 
781 
by (fact Inf_empty) (* already simp *) 

41082  782 

44067  783 
lemma Inter_UNIV: "\<Inter>UNIV = {}" 
784 
by (fact Inf_UNIV) (* already simp *) 

41082  785 

44920  786 
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" 
787 
by (fact Inf_insert) (* already simp *) 

41082  788 

789 
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" 

43899  790 
by (fact less_eq_Inf_inter) 
41082  791 

792 
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" 

43756  793 
by (fact Inf_union_distrib) 
794 

43868  795 
lemma Inter_UNIV_conv [simp, no_atp]: 
43741  796 
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" 
797 
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" 

43801  798 
by (fact Inf_top_conv)+ 
41082  799 

43741  800 
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" 
43899  801 
by (fact Inf_superset_mono) 
41082  802 

803 

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

804 
subsubsection {* Intersections of families *} 
41082  805 

806 
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 

807 
"INTER \<equiv> INFI" 

808 

43872  809 
text {* 
810 
Note: must use name @{const INTER} here instead of @{text INT} 

811 
to allow the following syntax coexist with the plain constant name. 

812 
*} 

813 

41082  814 
syntax 
815 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

816 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) 

817 

818 
syntax (xsymbols) 

819 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

820 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) 

821 

822 
syntax (latex output) 

823 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

824 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) 

825 

826 
translations 

827 
"INT x y. B" == "INT x. INT y. B" 

828 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

829 
"INT x. B" == "INT x:CONST UNIV. B" 

830 
"INT x:A. B" == "CONST INTER A (%x. B)" 

831 

832 
print_translation {* 

42284  833 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] 
41082  834 
*}  {* to avoid etacontraction of body *} 
835 

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

836 
lemma INTER_eq: 
41082  837 
"(\<Inter>x\<in>A. B x) = {y. \<forall>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

838 
by (auto simp add: INF_def) 
41082  839 

840 
lemma Inter_image_eq [simp]: 

841 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 

43872  842 
by (rule sym) (fact INF_def) 
41082  843 

43817  844 
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

845 
by (auto simp add: INF_def image_def) 
41082  846 

43817  847 
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

848 
by (auto simp add: INF_def image_def) 
41082  849 

43852  850 
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" 
41082  851 
by auto 
852 

43852  853 
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" 
854 
 {* "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

855 
by (auto simp add: INF_def image_def) 
41082  856 

857 
lemma INT_cong [cong]: 

43854  858 
"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

859 
by (fact INF_cong) 
41082  860 

861 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 

862 
by blast 

863 

864 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 

865 
by blast 

866 

43817  867 
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

868 
by (fact INF_lower) 
41082  869 

43817  870 
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

871 
by (fact INF_greatest) 
41082  872 

44067  873 
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

874 
by (fact INF_empty) 
43854  875 

43817  876 
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" 
43872  877 
by (fact INF_absorb) 
41082  878 

43854  879 
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" 
41082  880 
by (fact le_INF_iff) 
881 

882 
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

883 
by (fact INF_insert) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

884 

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

885 
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

886 
by (fact INF_union) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

887 

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

888 
lemma INT_insert_distrib: 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

889 
"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

890 
by blast 
43854  891 

41082  892 
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

893 
by (fact INF_constant) 
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

894 

44920  895 
lemma INTER_UNIV_conv: 
43817  896 
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" 
897 
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" 

44920  898 
by (fact INF_top_conv)+ (* already simp *) 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

899 

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

900 
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" 
43873  901 
by (fact INF_UNIV_bool_expand) 
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset

902 

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

903 
lemma INT_anti_mono: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

904 
"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

905 
 {* The last inclusion is POSITIVE! *} 
43940  906 
by (fact INF_superset_mono) 
41082  907 

908 
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" 

909 
by blast 

910 

43817  911 
lemma vimage_INT: "f ` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f ` B x)" 
41082  912 
by blast 
913 

914 

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

915 
subsubsection {* Union *} 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

916 

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

917 
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

918 
"Union S \<equiv> \<Squnion>S" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

919 

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

920 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

921 
Union ("\<Union>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

922 

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

923 
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

924 
"\<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

925 
proof (rule set_eqI) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

926 
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

927 
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

928 
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

929 
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

930 
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

931 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

932 

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

933 
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

934 
"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

935 
by (unfold Union_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

936 

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

937 
lemma UnionI [intro]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

938 
"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

939 
 {* 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

940 
@{term A} may be flexible. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

941 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

942 

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

943 
lemma UnionE [elim!]: 
43817  944 
"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

945 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

946 

43817  947 
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" 
43901  948 
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

949 

43817  950 
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" 
43901  951 
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

952 

44920  953 
lemma Union_empty: "\<Union>{} = {}" 
954 
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

955 

44920  956 
lemma Union_UNIV: "\<Union>UNIV = UNIV" 
957 
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

958 

44920  959 
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B" 
960 
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

961 

43817  962 
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" 
43901  963 
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

964 

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

965 
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" 
43901  966 
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

967 

44920  968 
lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" 
969 
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

970 

44920  971 
lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" 
972 
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

973 

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

974 
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

975 
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

976 

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

977 
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

978 
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

979 

43817  980 
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" 
43901  981 
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

982 

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

983 

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

984 
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

985 

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

986 
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

987 
"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

988 

43872  989 
text {* 
990 
Note: must use name @{const UNION} here instead of @{text UN} 

991 
to allow the following syntax coexist with the plain constant name. 

992 
*} 

993 

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

994 
syntax 
35115  995 
"_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

996 
"_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

997 

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

998 
syntax (xsymbols) 
35115  999 
"_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

1000 
"_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

1001 

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

1002 
syntax (latex output) 
35115  1003 
"_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

1004 
"_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

1005 

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 
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

1007 
"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

1008 
"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

1009 
"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

1010 
"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

1011 

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

1012 
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

1013 
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

1014 
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

1015 
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

1016 
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

1017 
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

1018 
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

1019 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

1020 

35115  1021 
print_translation {* 
42284  1022 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] 
35115  1023 
*}  {* 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

1024 

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

1025 
lemma UNION_eq [no_atp]: 
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

1026 
"(\<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

1027 
by (auto simp add: SUP_def) 
44920  1028 

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

1029 
lemma bind_UNION [code]: 
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset

1030 
"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

1031 
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

1032 

46036  1033 
lemma member_bind [simp]: 
1034 
"x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f " 

1035 
by (simp add: bind_UNION) 

1036 

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

1037 
lemma Union_image_eq [simp]: 
43817  1038 
"\<Union>(B ` A) = (\<Union>x\<in>A. B x)" 
44920  1039 
by (rule sym) (fact SUP_def) 
1040 

46036  1041 
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

1042 
by (auto simp add: SUP_def image_def) 
11979  1043 

43852  1044 
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" 
11979  1045 
 {* The order of the premises presupposes that @{term A} is rigid; 
1046 
@{term b} may be flexible. *} 

1047 
by auto 

1048 

43852  1049 
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

1050 
by (auto simp add: SUP_def image_def) 
923  1051 

11979  1052 
lemma UN_cong [cong]: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1053 
"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

1054 
by (fact SUP_cong) 
11979  1055 

29691  1056 
lemma strong_UN_cong: 
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset

1057 
"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

1058 
by (unfold simp_implies_def) (fact UN_cong) 
29691  1059 

43817  1060 
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

1061 
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

1062 

43817  1063 
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

1064 
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

1065 

43817  1066 
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

1067 
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

1068 

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

1069 
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

1070 
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

1071 

43817  1072 
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

1073 
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

1074 

44067  1075 
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

1076 
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

1077 

44920  1078 
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" 
1079 
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

1080 

43817  1081 
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

1082 
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

1083 

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

1084 
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

1085 
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

1086 

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

1087 
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

1088 
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

1089 

43967  1090 
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

1091 
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

1092 

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

1093 
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" 
35629  1094 
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

1095 

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

1096 
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

1097 
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

1098 

43944  1099 
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

1100 
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

1101 

44920  1102 
lemma UNION_empty_conv: 
43817  1103 
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" 
1104 
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" 

44920  1105 
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

1106 

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

1107 
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

1108 
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

1109 

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

1110 
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

1111 
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

1112 

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

1113 
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

1114 
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

1115 

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

1116 
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

1117 
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

1118 

43817  1119 
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

1120 
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

1121 

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

1122 
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

1123 
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

1124 

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

1125 
lemma UN_mono: 
43817  1126 
"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

1127 
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" 
43940  1128 
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

1129 

43817  1130 
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

1131 
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

1132 

43817  1133 
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

1134 
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

1135 

43817  1136 
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

1137 
 {* 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

1138 
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

1139 

43817  1140 
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" 
1141 
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

1142 

45013  1143 
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" 
1144 
by blast 

1145 

11979  1146 

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

1147 
subsubsection {* Distributive laws *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1148 

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

1149 
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

1150 
by (fact inf_Sup) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1151 

44039  1152 
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" 
1153 
by (fact sup_Inf) 

1154 

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

1155 
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" 
44039  1156 
by (fact Sup_inf) 
1157 

1158 
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)" 

1159 
by (rule sym) (rule INF_inf_distrib) 

1160 

1161 
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)" 

1162 
by (rule sym) (rule SUP_sup_distrib) 

1163 

1164 
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" 

1165 
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

1166 

43817  1167 
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

1168 
 {* 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

1169 
 {* Union of a family of unions *} 
44039  1170 
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

1171 

44039  1172 
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" 
1173 
by (fact sup_INF) 

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

1174 

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

1175 
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

1176 
 {* Halmos, Naive Set Theory, page 35. *} 
44039  1177 
by (fact inf_SUP) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1178 

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

1179 
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  1180 
by (fact SUP_inf_distrib2) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1181 

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

1182 
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  1183 
by (fact INF_sup_distrib2) 
1184 

1185 
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" 

1186 
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

1187 

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

1188 

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

1189 
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

1190 

43873  1191 
lemma Compl_INT [simp]: " (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. B x)" 
1192 
by (fact uminus_INF) 

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

1193 

43873  1194 
lemma Compl_UN [simp]: " (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. B x)" 
1195 
by (fact uminus_SUP) 

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

1196 

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

1197 

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

1198 
subsubsection {* Miniscoping and maxiscoping *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1199 

13860  1200 
text {* \medskip Miniscoping: pushing in quantifiers and big Unions 
1201 
and Intersections. *} 

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

1202 

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

1203 
lemma UN_simps [simp]: 
43817  1204 
"\<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

1205 
"\<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  1206 
"\<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

1207 
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" 
43852  1208 
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" 
1209 
"\<And>A B C. (\<Union>x\<in>C. A x  B) = ((\<Union>x\<in>C. A x)  B)" 

1210 
"\<And>A B C. (\<Union>x\<in>C. A  B x) = (A  (\<Inter>x\<in>C. B x))" 

1211 
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" 

1212 
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" 

43831  1213 
"\<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

1214 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1215 

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

1216 
lemma INT_simps [simp]: 
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset

1217 
"\<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 