author  huffman 
Sat, 06 Mar 2010 08:08:30 0800  
changeset 35629  57f1a5e93b6b 
parent 35115  446c5063e4fd 
child 35828  46cfc4b8112e 
permissions  rwrr 
32139  1 
(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) 
11979  2 

32139  3 
header {* Complete lattices, with special focus on sets *} 
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 

32139  5 
theory Complete_Lattice 
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 
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

11 
less (infix "\<sqsubset>" 50) and 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

12 
inf (infixl "\<sqinter>" 70) and 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

13 
sup (infixl "\<squnion>" 65) and 
32678  14 
top ("\<top>") and 
15 
bot ("\<bottom>") 

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

16 

32139  17 

32879  18 
subsection {* Syntactic infimum and supremum operations *} 
19 

20 
class Inf = 

21 
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 

22 

23 
class Sup = 

24 
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 

25 

32139  26 
subsection {* Abstract complete lattices *} 
27 

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

28 
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

29 
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

30 
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

31 
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

32 
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

33 
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

34 

32678  35 
lemma dual_complete_lattice: 
32879  36 
"complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>" 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

37 
by (auto intro!: complete_lattice.intro dual_bounded_lattice) 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

38 
(unfold_locales, (fact bot_least top_greatest 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

39 
Sup_upper Sup_least Inf_lower Inf_greatest)+) 
32678  40 

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

41 
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> 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

42 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

43 

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

44 
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" 
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

45 
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

46 

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

47 
lemma Inf_empty: 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

48 
"\<Sqinter>{} = \<top>" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

49 
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

50 

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

51 
lemma Sup_empty: 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

52 
"\<Squnion>{} = \<bottom>" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

53 
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

54 

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

55 
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<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

56 
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

57 

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

58 
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<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

59 
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

60 

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

61 
lemma Inf_singleton [simp]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

62 
"\<Sqinter>{a} = 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

63 
by (auto intro: antisym Inf_lower Inf_greatest) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

64 

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

65 
lemma Sup_singleton [simp]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

66 
"\<Squnion>{a} = 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

67 
by (auto intro: antisym Sup_upper Sup_least) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

68 

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

69 
lemma Inf_binary: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

70 
"\<Sqinter>{a, b} = a \<sqinter> b" 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

71 
by (simp add: Inf_empty Inf_insert) 
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

72 

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

73 
lemma Sup_binary: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

74 
"\<Squnion>{a, b} = a \<squnion> b" 
34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

75 
by (simp add: Sup_empty Sup_insert) 
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

76 

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

77 
lemma Inf_UNIV: 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

78 
"\<Sqinter>UNIV = bot" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

79 
by (simp add: Sup_Inf Sup_empty [symmetric]) 
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

80 

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

81 
lemma Sup_UNIV: 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

82 
"\<Squnion>UNIV = top" 
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
32879
diff
changeset

83 
by (simp add: Inf_Sup Inf_empty [symmetric]) 
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

84 

35629  85 
lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" 
86 
by (auto intro: Sup_least dest: Sup_upper) 

87 

88 
lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" 

89 
by (auto intro: Inf_greatest dest: Inf_lower) 

90 

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

91 
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

92 
"SUPR A f = \<Squnion> (f ` 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

93 

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

94 
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

95 
"INFI A f = \<Sqinter> (f ` 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

96 

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

97 
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

98 

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

99 
syntax 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

100 
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

101 
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

102 
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

103 
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

104 

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

105 
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

106 
"SUP x y. B" == "SUP x. SUP 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

107 
"SUP x. B" == "CONST SUPR 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

108 
"SUP x. B" == "SUP 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

109 
"SUP x:A. B" == "CONST SUPR 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

110 
"INF x y. B" == "INF x. INF 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

111 
"INF x. B" == "CONST INFI 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

112 
"INF x. B" == "INF 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

113 
"INF x:A. B" == "CONST INFI 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

114 

35115  115 
print_translation {* 
116 
[Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}, 

117 
Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}] 

118 
*}  {* to avoid etacontraction of body *} 

11979  119 

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

120 
context 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

121 
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

122 

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

123 
lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)" 
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

124 
by (auto simp add: SUPR_def intro: 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

125 

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

126 
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u" 
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

127 
by (auto simp add: SUPR_def intro: Sup_least) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

128 

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

129 
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i" 
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

130 
by (auto simp add: INFI_def intro: 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

131 

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

132 
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)" 
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

133 
by (auto simp add: INFI_def intro: Inf_greatest) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

134 

35629  135 
lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)" 
136 
unfolding SUPR_def by (auto simp add: Sup_le_iff) 

137 

138 
lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)" 

139 
unfolding INFI_def by (auto simp add: le_Inf_iff) 

140 

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

141 
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

142 
by (auto intro: antisym SUP_leI le_SUPI) 
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 

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

144 
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

145 
by (auto intro: antisym INF_leI le_INFI) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

146 

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 
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

148 

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

149 

32139  150 
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as 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

151 

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

152 
instantiation bool :: 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

153 
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

154 

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

155 
definition 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

156 
Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. 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

157 

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

158 
definition 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

159 
Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. 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

160 

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

161 
instance proof 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

162 
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

163 

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

164 
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

165 

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

166 
lemma Inf_empty_bool [simp]: 
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 
"\<Sqinter>{}" 
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 
unfolding Inf_bool_def by auto 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

169 

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

170 
lemma not_Sup_empty_bool [simp]: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

171 
"\<not> \<Squnion>{}" 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

172 
unfolding Sup_bool_def by auto 
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 

32120
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

174 
lemma INFI_bool_eq: 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

175 
"INFI = Ball" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

176 
proof (rule ext)+ 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

177 
fix A :: "'a set" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

178 
fix P :: "'a \<Rightarrow> bool" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

179 
show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

180 
by (auto simp add: Ball_def INFI_def Inf_bool_def) 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

181 
qed 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

182 

53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

183 
lemma SUPR_bool_eq: 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

184 
"SUPR = Bex" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

185 
proof (rule ext)+ 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

186 
fix A :: "'a set" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

187 
fix P :: "'a \<Rightarrow> bool" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

188 
show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)" 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

189 
by (auto simp add: Bex_def SUPR_def Sup_bool_def) 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

190 
qed 
53a21a5e6889
attempt for more concise setup of nonetacontracting binders
haftmann
parents:
32117
diff
changeset

191 

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

192 
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

193 
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

194 

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

195 
definition 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

196 
Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f 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

197 

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

198 
definition 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

199 
Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f 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

200 

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

201 
instance proof 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

202 
qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

203 
intro: Inf_lower Sup_upper Inf_greatest Sup_least) 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

204 

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

205 
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

206 

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

207 
lemma Inf_empty_fun: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

208 
"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" 
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

209 
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

210 

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

211 
lemma Sup_empty_fun: 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

212 
"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" 
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

213 
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

214 

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

215 

32139  216 
subsection {* Union *} 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

217 

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

218 
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

219 
"Union S \<equiv> \<Squnion>S" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

220 

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

221 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

222 
Union ("\<Union>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

223 

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

224 
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

225 
"\<Union>A = {x. \<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

226 
proof (rule set_ext) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

227 
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

228 
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

229 
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

230 
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" 
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

231 
by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

232 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

233 

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

234 
lemma Union_iff [simp, noatp]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

235 
"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

236 
by (unfold Union_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

237 

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

238 
lemma UnionI [intro]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

239 
"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

240 
 {* 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

241 
@{term A} may be flexible. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

242 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

243 

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

244 
lemma UnionE [elim!]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

245 
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

246 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

247 

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

248 
lemma Union_upper: "B \<in> A ==> B \<subseteq> 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

249 
by (iprover intro: subsetI UnionI) 
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

250 

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

251 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
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

252 
by (iprover intro: subsetI elim: UnionE dest: subsetD) 
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

253 

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

254 
lemma Un_eq_Union: "A \<union> B = \<Union>{A, 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

255 
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

256 

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

257 
lemma Union_empty [simp]: "Union({}) = {}" 
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

258 
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

259 

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

260 
lemma Union_UNIV [simp]: "Union UNIV = UNIV" 
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

261 
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

262 

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

263 
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>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

264 
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

265 

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

266 
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>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

267 
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

268 

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

269 
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>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

270 
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

271 

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

272 
lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. 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

273 
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

274 

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

275 
lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. 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

276 
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

277 

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

278 
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> 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

279 
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

280 

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

281 
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

282 
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

283 

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

284 
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

285 
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

286 

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

287 
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>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

288 
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

289 

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

290 

32139  291 
subsection {* 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

292 

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

293 
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

294 
"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

295 

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

296 
syntax 
35115  297 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 
298 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [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

299 

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

300 
syntax (xsymbols) 
35115  301 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 
302 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [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

303 

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

304 
syntax (latex output) 
35115  305 
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
306 
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [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

307 

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

308 
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

309 
"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

310 
"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

311 
"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

312 
"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

313 

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

314 
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

315 
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

316 
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

317 
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

318 
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

319 
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

320 
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

321 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

322 

35115  323 
print_translation {* 
324 
[Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] 

325 
*}  {* 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

326 

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

327 
lemma UNION_eq_Union_image: 
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

328 
"(\<Union>x\<in>A. B x) = \<Union>(B`A)" 
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

329 
by (fact SUPR_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

330 

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

331 
lemma Union_def: 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

332 
"\<Union>S = (\<Union>x\<in>S. x)" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

333 
by (simp add: UNION_eq_Union_image image_def) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

334 

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

335 
lemma UNION_def [noatp]: 
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

336 
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

337 
by (auto simp add: UNION_eq_Union_image Union_eq) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

338 

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

339 
lemma Union_image_eq [simp]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

340 
"\<Union>(B`A) = (\<Union>x\<in>A. B x)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

341 
by (rule sym) (fact UNION_eq_Union_image) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

342 

11979  343 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
344 
by (unfold UNION_def) blast 

345 

346 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

347 
 {* The order of the premises presupposes that @{term A} is rigid; 

348 
@{term b} may be flexible. *} 

349 
by auto 

350 

351 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

352 
by (unfold UNION_def) blast 

923  353 

11979  354 
lemma UN_cong [cong]: 
355 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

356 
by (simp add: UNION_def) 

357 

29691  358 
lemma strong_UN_cong: 
359 
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

360 
by (simp add: UNION_def simp_implies_def) 

361 

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

362 
lemma image_eq_UN: "f`A = (UN x:A. {f 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

363 
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

364 

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

365 
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)" 
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

366 
by (fact le_SUPI) 
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

367 

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

368 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
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

369 
by (iprover intro: subsetI elim: UN_E dest: subsetD) 
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

370 

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

371 
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
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

372 
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

373 

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

374 
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<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

375 
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

376 

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

377 
lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. 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

378 
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

379 

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

380 
lemma UN_empty2 [simp]: "(\<Union>x\<in>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

381 
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

382 

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

383 
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = 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

384 
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

385 

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

386 
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" 
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

387 
by auto 
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

388 

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

389 
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A 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

390 
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

391 

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

392 
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)" 
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

393 
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

394 

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

395 
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)" 
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

396 
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

397 

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

398 
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" 
35629  399 
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

400 

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

401 
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` 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

402 
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

403 

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

404 
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" 
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

405 
by auto 
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

406 

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

407 
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = 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

408 
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

409 

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

410 
lemma UNION_empty_conv[simp]: 
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

411 
"({} = (UN x:A. B x)) = (\<forall>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

412 
"((UN x:A. B x) = {}) = (\<forall>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

413 
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

414 

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

415 
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
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

416 
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

417 

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

418 
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" 
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

419 
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

420 

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

421 
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" 
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

422 
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

423 

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

424 
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

425 
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

426 

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

427 
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)" 
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

428 
by (auto intro: bool_contrapos) 
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

429 

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

430 
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

431 
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

432 

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

433 
lemma UN_mono: 
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

434 
"A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g 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

435 
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g 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

436 
by (blast dest: subsetD) 
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

437 

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

438 
lemma vimage_Union: "f ` (Union A) = (UN X:A. f ` 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

439 
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

440 

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

441 
lemma vimage_UN: "f`(UN x:A. B x) = (UN x:A. f ` 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

442 
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

443 

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

444 
lemma vimage_eq_UN: "f`B = (UN y: B. f`{y})" 
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

445 
 {* 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

446 
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

447 

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

448 
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (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

449 
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

450 

11979  451 

32139  452 
subsection {* Inter *} 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

453 

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

454 
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where 
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

455 
"Inter S \<equiv> \<Sqinter>S" 
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

456 

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

457 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

458 
Inter ("\<Inter>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

459 

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

460 
lemma Inter_eq [code del]: 
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

461 
"\<Inter>A = {x. \<forall>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

462 
proof (rule set_ext) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

463 
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

464 
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>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

465 
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

466 
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" 
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset

467 
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

468 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

469 

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

470 
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

471 
by (unfold Inter_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

472 

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

473 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

474 
by (simp add: Inter_eq) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

475 

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

476 
text {* 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

477 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

478 
contains @{term A} as an element, but @{prop "A:X"} can hold when 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

479 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

480 
*} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

481 

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

482 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

483 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

484 

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

485 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

486 
 {* ``Classical'' elimination rule  does not require proving 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

487 
@{prop "X:C"}. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

488 
by (unfold Inter_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

489 

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

490 
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> 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

491 
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

492 

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

493 
lemma Inter_subset: 
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

494 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> 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

495 
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

496 

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

497 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter 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

498 
by (iprover intro: InterI subsetI dest: subsetD) 
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

499 

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

500 
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, 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

501 
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

502 

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

503 
lemma Inter_empty [simp]: "\<Inter>{} = UNIV" 
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

504 
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

505 

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

506 
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}" 
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

507 
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

508 

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

509 
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>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

510 
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

511 

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

512 
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> 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

513 
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

514 

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

515 
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>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

516 
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

517 

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

518 
lemma Inter_UNIV_conv [simp,noatp]: 
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

519 
"(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)" 
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

520 
"(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)" 
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

521 
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

522 

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

523 
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>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

524 
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

525 

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

526 

32139  527 
subsection {* Intersections of families *} 
11979  528 

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

529 
abbreviation INTER :: "'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

530 
"INTER \<equiv> INFI" 
32081  531 

532 
syntax 

35115  533 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 
534 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) 

32081  535 

536 
syntax (xsymbols) 

35115  537 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 
538 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) 

32081  539 

540 
syntax (latex output) 

35115  541 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
542 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

32081  543 

544 
translations 

545 
"INT x y. B" == "INT x. INT y. B" 

546 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

547 
"INT x. B" == "INT x:CONST UNIV. B" 

548 
"INT x:A. B" == "CONST INTER A (%x. B)" 

549 

35115  550 
print_translation {* 
551 
[Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] 

552 
*}  {* to avoid etacontraction of body *} 

32081  553 

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

554 
lemma INTER_eq_Inter_image: 
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

555 
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)" 
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

556 
by (fact INFI_def) 
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

557 

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

558 
lemma Inter_def: 
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

559 
"\<Inter>S = (\<Inter>x\<in>S. x)" 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

560 
by (simp add: INTER_eq_Inter_image image_def) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

561 

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

562 
lemma INTER_def: 
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

563 
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

564 
by (auto simp add: INTER_eq_Inter_image Inter_eq) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

565 

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

566 
lemma Inter_image_eq [simp]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

567 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

568 
by (rule sym) (fact INTER_eq_Inter_image) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

569 

11979  570 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 
571 
by (unfold INTER_def) blast 

923  572 

11979  573 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
574 
by (unfold INTER_def) blast 

575 

576 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

577 
by auto 

578 

579 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

580 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

581 
by (unfold INTER_def) blast 

582 

583 
lemma INT_cong [cong]: 

584 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

585 
by (simp add: INTER_def) 

7238
36e58620ffc8
replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents:
5931
diff
changeset

586 

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

587 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 
30531
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

588 
by blast 
ab3d61baf66a
reverted to old version of Set.thy  strange effects have to be traced first
haftmann
parents:
30352
diff
changeset

589 

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

590 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

591 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

592 

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

593 
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a" 
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

594 
by (fact INF_leI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

595 

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

596 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

597 
by (fact le_INFI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

598 

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

599 
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

600 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

601 

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

602 
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

603 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

604 

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

605 
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)" 
35629  606 
by (fact le_INF_iff) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

607 

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

608 
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

609 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

610 

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

611 
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

612 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

613 

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

614 
lemma INT_insert_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

615 
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

616 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

617 

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

618 
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

619 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

620 

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

621 
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

622 
 {* Look: it has an \emph{existential} quantifier *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

623 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

624 

18447  625 
lemma INTER_UNIV_conv[simp]: 
13653  626 
"(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)" 
627 
"((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" 

628 
by blast+ 

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

629 

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

630 
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)" 
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

631 
by (auto intro: bool_induct) 
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

632 

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

633 
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (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

634 
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

635 

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

636 
lemma INT_anti_mono: 
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

637 
"B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g 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

638 
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g 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

639 
 {* The last inclusion is POSITIVE! *} 
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

640 
by (blast dest: subsetD) 
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

641 

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

642 
lemma vimage_INT: "f`(INT x:A. B x) = (INT x:A. f ` 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

643 
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

644 

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

645 

32139  646 
subsection {* Distributive laws *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

647 

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

648 
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

649 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

650 

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

651 
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

652 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

653 

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

654 
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

655 
 {* 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

656 
 {* Union of a family of unions *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

657 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

658 

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

659 
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)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

660 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

661 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

662 

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

663 
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

664 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

665 

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

666 
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

667 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

668 

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

669 
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)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

670 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

671 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

672 

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

673 
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

674 
 {* Halmos, Naive Set Theory, page 35. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

675 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

676 

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

677 
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

678 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

679 

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

680 
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)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

681 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

682 

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

683 
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)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

684 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

685 

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

686 

32139  687 
subsection {* 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

688 

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

689 
lemma Compl_UN [simp]: "(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. B x)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

690 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

691 

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

692 
lemma Compl_INT [simp]: "(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. B x)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

693 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

694 

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

695 

32139  696 
subsection {* Miniscoping and maxiscoping *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

697 

13860  698 
text {* \medskip Miniscoping: pushing in quantifiers and big Unions 
699 
and Intersections. *} 

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

700 

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

701 
lemma UN_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

702 
"!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

703 
"!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

704 
"!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

705 
"!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

706 
"!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

707 
"!!A B C. (UN x:C. A x  B) = ((UN x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

708 
"!!A B C. (UN x:C. A  B x) = (A  (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

709 
"!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

710 
"!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

711 
"!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

712 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

713 

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

714 
lemma INT_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

715 
"!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

716 
"!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

717 
"!!A B C. (INT x:C. A x  B) = (if C={} then UNIV else (INT x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

718 
"!!A B C. (INT x:C. A  B x) = (if C={} then UNIV else A  (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

719 
"!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

720 
"!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

721 
"!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

722 
"!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

723 
"!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

724 
"!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

725 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

726 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

727 
lemma ball_simps [simp,noatp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

728 
"!!A P Q. (ALL x:A. P x  Q) = ((ALL x:A. P x)  Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

729 
"!!A P Q. (ALL x:A. P  Q x) = (P  (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

730 
"!!A P Q. (ALL x:A. P > Q x) = (P > (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

731 
"!!A P Q. (ALL x:A. P x > Q) = ((EX x:A. P x) > Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

732 
"!!P. (ALL x:{}. P x) = True" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

733 
"!!P. (ALL x:UNIV. P x) = (ALL x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

734 
"!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

735 
"!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

736 
"!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

737 
"!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x > P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

738 
"!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

739 
"!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

740 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

741 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

742 
lemma bex_simps [simp,noatp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

743 
"!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

744 
"!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

745 
"!!P. (EX x:{}. P x) = False" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

746 
"!!P. (EX x:UNIV. P x) = (EX x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

747 
"!!a B P. (EX x:insert a B. P x) = (P(a)  (EX x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

748 
"!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

749 
"!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

750 
"!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

751 
"!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

752 
"!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

753 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

754 

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

755 
lemma ball_conj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

756 
"(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

757 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

758 

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

759 
lemma bex_disj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

760 
"(EX x:A. P x  Q x) = ((EX x:A. P x)  (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

761 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

762 

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

763 

13860  764 
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} 
765 

766 
lemma UN_extend_simps: 

767 
"!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))" 

768 
"!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))" 

769 
"!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))" 

770 
"!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)" 

771 
"!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)" 

772 
"!!A B C. ((UN x:C. A x)  B) = (UN x:C. A x  B)" 

773 
"!!A B C. (A  (INT x:C. B x)) = (UN x:C. A  B x)" 

774 
"!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)" 

775 
"!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)" 

776 
"!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)" 

777 
by auto 

778 

779 
lemma INT_extend_simps: 

780 
"!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))" 

781 
"!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))" 

782 
"!!A B C. (INT x:C. A x)  B = (if C={} then UNIVB else (INT x:C. A x  B))" 

783 
"!!A B C. A  (UN x:C. B x) = (if C={} then A else (INT x:C. A  B x))" 

784 
"!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))" 

785 
"!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)" 

786 
"!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)" 

787 
"!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)" 

788 
"!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)" 

789 
"!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)" 

790 
by auto 

791 

792 

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

793 
no_notation 
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

794 
less_eq (infix "\<sqsubseteq>" 50) and 
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

795 
less (infix "\<sqsubset>" 50) and 
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

796 
inf (infixl "\<sqinter>" 70) and 
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

797 
sup (infixl "\<squnion>" 65) and 
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

798 
Inf ("\<Sqinter>_" [900] 900) and 
32678  799 
Sup ("\<Squnion>_" [900] 900) and 
800 
top ("\<top>") and 

801 
bot ("\<bottom>") 

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

802 

30596  803 
lemmas mem_simps = 
804 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

805 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

806 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

21669  807 

11979  808 
end 