src/HOL/Complete_Lattice.thy
author haftmann
Sun Jul 10 22:17:33 2011 +0200 (2011-07-10)
changeset 43755 5e4a595e63fc
parent 43754 09d455c93bf8
child 43756 15ea1a07a8cf
permissions -rw-r--r--
tuned notation
haftmann@32139
     1
(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
wenzelm@11979
     2
haftmann@32139
     3
header {* Complete lattices, with special focus on sets *}
haftmann@32077
     4
haftmann@32139
     5
theory Complete_Lattice
haftmann@32139
     6
imports Set
haftmann@32139
     7
begin
haftmann@32077
     8
haftmann@32077
     9
notation
haftmann@34007
    10
  less_eq (infix "\<sqsubseteq>" 50) and
haftmann@32077
    11
  less (infix "\<sqsubset>" 50) and
haftmann@34007
    12
  inf (infixl "\<sqinter>" 70) and
haftmann@34007
    13
  sup (infixl "\<squnion>" 65) and
haftmann@32678
    14
  top ("\<top>") and
haftmann@32678
    15
  bot ("\<bottom>")
haftmann@32077
    16
haftmann@32139
    17
haftmann@32879
    18
subsection {* Syntactic infimum and supremum operations *}
haftmann@32879
    19
haftmann@32879
    20
class Inf =
haftmann@32879
    21
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
haftmann@32879
    22
haftmann@32879
    23
class Sup =
haftmann@32879
    24
  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
haftmann@32879
    25
haftmann@32139
    26
subsection {* Abstract complete lattices *}
haftmann@32139
    27
haftmann@34007
    28
class complete_lattice = bounded_lattice + Inf + Sup +
haftmann@32077
    29
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
haftmann@32077
    30
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
haftmann@32077
    31
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
haftmann@32077
    32
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
haftmann@32077
    33
begin
haftmann@32077
    34
haftmann@32678
    35
lemma dual_complete_lattice:
haftmann@36635
    36
  "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
haftmann@36635
    37
  by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
haftmann@34007
    38
    (unfold_locales, (fact bot_least top_greatest
haftmann@34007
    39
        Sup_upper Sup_least Inf_lower Inf_greatest)+)
haftmann@32678
    40
haftmann@34007
    41
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
haftmann@32077
    42
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@32077
    43
haftmann@34007
    44
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
haftmann@32077
    45
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@32077
    46
haftmann@41080
    47
lemma Inf_empty [simp]:
haftmann@34007
    48
  "\<Sqinter>{} = \<top>"
haftmann@34007
    49
  by (auto intro: antisym Inf_greatest)
haftmann@32077
    50
haftmann@41080
    51
lemma Sup_empty [simp]:
haftmann@34007
    52
  "\<Squnion>{} = \<bottom>"
haftmann@34007
    53
  by (auto intro: antisym Sup_least)
haftmann@32077
    54
haftmann@41080
    55
lemma Inf_UNIV [simp]:
haftmann@41080
    56
  "\<Sqinter>UNIV = \<bottom>"
haftmann@41080
    57
  by (simp add: Sup_Inf Sup_empty [symmetric])
haftmann@41080
    58
haftmann@41080
    59
lemma Sup_UNIV [simp]:
haftmann@41080
    60
  "\<Squnion>UNIV = \<top>"
haftmann@41080
    61
  by (simp add: Inf_Sup Inf_empty [symmetric])
haftmann@41080
    62
haftmann@32077
    63
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
haftmann@32077
    64
  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
haftmann@32077
    65
haftmann@32077
    66
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
haftmann@32077
    67
  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
haftmann@32077
    68
haftmann@32077
    69
lemma Inf_singleton [simp]:
haftmann@32077
    70
  "\<Sqinter>{a} = a"
haftmann@32077
    71
  by (auto intro: antisym Inf_lower Inf_greatest)
haftmann@32077
    72
haftmann@32077
    73
lemma Sup_singleton [simp]:
haftmann@32077
    74
  "\<Squnion>{a} = a"
haftmann@32077
    75
  by (auto intro: antisym Sup_upper Sup_least)
haftmann@32077
    76
haftmann@32077
    77
lemma Inf_binary:
haftmann@32077
    78
  "\<Sqinter>{a, b} = a \<sqinter> b"
haftmann@34007
    79
  by (simp add: Inf_empty Inf_insert)
haftmann@32077
    80
haftmann@32077
    81
lemma Sup_binary:
haftmann@32077
    82
  "\<Squnion>{a, b} = a \<squnion> b"
haftmann@34007
    83
  by (simp add: Sup_empty Sup_insert)
haftmann@32077
    84
haftmann@43754
    85
lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
huffman@35629
    86
  by (auto intro: Inf_greatest dest: Inf_lower)
huffman@35629
    87
haftmann@43741
    88
lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
haftmann@41082
    89
  by (auto intro: Sup_least dest: Sup_upper)
hoelzl@38705
    90
hoelzl@38705
    91
lemma Inf_mono:
hoelzl@41971
    92
  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
haftmann@43741
    93
  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
hoelzl@38705
    94
proof (rule Inf_greatest)
hoelzl@38705
    95
  fix b assume "b \<in> B"
hoelzl@41971
    96
  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
haftmann@43741
    97
  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
haftmann@43741
    98
  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
hoelzl@38705
    99
qed
hoelzl@38705
   100
haftmann@41082
   101
lemma Sup_mono:
hoelzl@41971
   102
  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
haftmann@43741
   103
  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
haftmann@41082
   104
proof (rule Sup_least)
haftmann@41082
   105
  fix a assume "a \<in> A"
hoelzl@41971
   106
  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
haftmann@43741
   107
  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
haftmann@43741
   108
  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
haftmann@41082
   109
qed
haftmann@32077
   110
hoelzl@41971
   111
lemma top_le:
haftmann@43741
   112
  "\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>"
hoelzl@41971
   113
  by (rule antisym) auto
hoelzl@41971
   114
hoelzl@41971
   115
lemma le_bot:
haftmann@43741
   116
  "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
hoelzl@41971
   117
  by (rule antisym) auto
hoelzl@41971
   118
haftmann@43741
   119
lemma not_less_bot[simp]: "\<not> (x \<sqsubset> \<bottom>)"
hoelzl@41971
   120
  using bot_least[of x] by (auto simp: le_less)
hoelzl@41971
   121
haftmann@43741
   122
lemma not_top_less[simp]: "\<not> (\<top> \<sqsubset> x)"
hoelzl@41971
   123
  using top_greatest[of x] by (auto simp: le_less)
hoelzl@41971
   124
haftmann@43741
   125
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
hoelzl@41971
   126
  using Sup_upper[of u A] by auto
hoelzl@41971
   127
haftmann@43741
   128
lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
hoelzl@41971
   129
  using Inf_lower[of u A] by auto
hoelzl@41971
   130
haftmann@32077
   131
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@32117
   132
  "INFI A f = \<Sqinter> (f ` A)"
haftmann@32077
   133
haftmann@41082
   134
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@41082
   135
  "SUPR A f = \<Squnion> (f ` A)"
haftmann@41082
   136
haftmann@32077
   137
end
haftmann@32077
   138
haftmann@32077
   139
syntax
haftmann@41082
   140
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@41082
   141
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
haftmann@41080
   142
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@41080
   143
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
haftmann@41080
   144
haftmann@41080
   145
syntax (xsymbols)
haftmann@41082
   146
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   147
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   148
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   149
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   150
haftmann@32077
   151
translations
haftmann@41082
   152
  "INF x y. B"   == "INF x. INF y. B"
haftmann@41082
   153
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
haftmann@41082
   154
  "INF x. B"     == "INF x:CONST UNIV. B"
haftmann@41082
   155
  "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@32077
   156
  "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@32077
   157
  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
haftmann@32077
   158
  "SUP x. B"     == "SUP x:CONST UNIV. B"
haftmann@32077
   159
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@32077
   160
wenzelm@35115
   161
print_translation {*
wenzelm@42284
   162
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
wenzelm@42284
   163
    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
wenzelm@35115
   164
*} -- {* to avoid eta-contraction of body *}
wenzelm@11979
   165
haftmann@32077
   166
context complete_lattice
haftmann@32077
   167
begin
haftmann@32077
   168
hoelzl@41971
   169
lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
hoelzl@41971
   170
  by (simp add: SUPR_def cong: image_cong)
hoelzl@41971
   171
hoelzl@41971
   172
lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
hoelzl@41971
   173
  by (simp add: INFI_def cong: image_cong)
hoelzl@41971
   174
haftmann@43741
   175
lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
haftmann@32077
   176
  by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@32077
   177
haftmann@43741
   178
lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
hoelzl@41971
   179
  using le_SUPI[of i A M] by auto
hoelzl@41971
   180
haftmann@43741
   181
lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"
haftmann@32077
   182
  by (auto simp add: SUPR_def intro: Sup_least)
haftmann@32077
   183
haftmann@43741
   184
lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"
haftmann@32077
   185
  by (auto simp add: INFI_def intro: Inf_lower)
haftmann@32077
   186
haftmann@43741
   187
lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"
hoelzl@41971
   188
  using INF_leI[of i A M] by auto
hoelzl@41971
   189
haftmann@43741
   190
lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"
haftmann@32077
   191
  by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@32077
   192
haftmann@43753
   193
lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
huffman@35629
   194
  unfolding SUPR_def by (auto simp add: Sup_le_iff)
huffman@35629
   195
haftmann@43753
   196
lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
huffman@35629
   197
  unfolding INFI_def by (auto simp add: le_Inf_iff)
huffman@35629
   198
haftmann@43753
   199
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"
haftmann@32077
   200
  by (auto intro: antisym INF_leI le_INFI)
haftmann@32077
   201
haftmann@43753
   202
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"
haftmann@41082
   203
  by (auto intro: antisym SUP_leI le_SUPI)
hoelzl@38705
   204
hoelzl@38705
   205
lemma INF_mono:
haftmann@43753
   206
  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
hoelzl@38705
   207
  by (force intro!: Inf_mono simp: INFI_def)
hoelzl@38705
   208
haftmann@41082
   209
lemma SUP_mono:
haftmann@43753
   210
  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
haftmann@41082
   211
  by (force intro!: Sup_mono simp: SUPR_def)
hoelzl@40872
   212
haftmann@43753
   213
lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
hoelzl@40872
   214
  by (intro INF_mono) auto
hoelzl@40872
   215
haftmann@43753
   216
lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
haftmann@41082
   217
  by (intro SUP_mono) auto
hoelzl@40872
   218
haftmann@43753
   219
lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
hoelzl@40872
   220
  by (iprover intro: INF_leI le_INFI order_trans antisym)
hoelzl@40872
   221
haftmann@43753
   222
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
haftmann@41082
   223
  by (iprover intro: SUP_leI le_SUPI order_trans antisym)
haftmann@41082
   224
haftmann@32077
   225
end
haftmann@32077
   226
haftmann@41082
   227
lemma Inf_less_iff:
haftmann@41082
   228
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753
   229
  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43754
   230
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@41082
   231
hoelzl@38705
   232
lemma less_Sup_iff:
hoelzl@38705
   233
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753
   234
  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43754
   235
  unfolding not_le [symmetric] Sup_le_iff by auto
hoelzl@38705
   236
haftmann@41082
   237
lemma INF_less_iff:
haftmann@41082
   238
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753
   239
  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@41082
   240
  unfolding INFI_def Inf_less_iff by auto
haftmann@32077
   241
hoelzl@40872
   242
lemma less_SUP_iff:
hoelzl@40872
   243
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753
   244
  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
hoelzl@40872
   245
  unfolding SUPR_def less_Sup_iff by auto
hoelzl@40872
   246
haftmann@32139
   247
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   248
haftmann@32077
   249
instantiation bool :: complete_lattice
haftmann@32077
   250
begin
haftmann@32077
   251
haftmann@32077
   252
definition
haftmann@41080
   253
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077
   254
haftmann@32077
   255
definition
haftmann@41080
   256
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077
   257
haftmann@32077
   258
instance proof
haftmann@32077
   259
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@32077
   260
haftmann@32077
   261
end
haftmann@32077
   262
haftmann@41080
   263
lemma INFI_bool_eq [simp]:
haftmann@32120
   264
  "INFI = Ball"
haftmann@32120
   265
proof (rule ext)+
haftmann@32120
   266
  fix A :: "'a set"
haftmann@32120
   267
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   268
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@32120
   269
    by (auto simp add: Ball_def INFI_def Inf_bool_def)
haftmann@32120
   270
qed
haftmann@32120
   271
haftmann@41080
   272
lemma SUPR_bool_eq [simp]:
haftmann@32120
   273
  "SUPR = Bex"
haftmann@32120
   274
proof (rule ext)+
haftmann@32120
   275
  fix A :: "'a set"
haftmann@32120
   276
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   277
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@32120
   278
    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
haftmann@32120
   279
qed
haftmann@32120
   280
haftmann@32077
   281
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   282
begin
haftmann@32077
   283
haftmann@32077
   284
definition
haftmann@41080
   285
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   286
haftmann@41080
   287
lemma Inf_apply:
haftmann@41080
   288
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   289
  by (simp add: Inf_fun_def)
haftmann@32077
   290
haftmann@32077
   291
definition
haftmann@41080
   292
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   293
haftmann@41080
   294
lemma Sup_apply:
haftmann@41080
   295
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   296
  by (simp add: Sup_fun_def)
haftmann@32077
   297
haftmann@32077
   298
instance proof
haftmann@41080
   299
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   300
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   301
haftmann@32077
   302
end
haftmann@32077
   303
haftmann@41080
   304
lemma INFI_apply:
haftmann@41080
   305
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@41080
   306
  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
hoelzl@38705
   307
haftmann@41080
   308
lemma SUPR_apply:
haftmann@41080
   309
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@41080
   310
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
haftmann@32077
   311
haftmann@32077
   312
haftmann@41082
   313
subsection {* Inter *}
haftmann@41082
   314
haftmann@41082
   315
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   316
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   317
  
haftmann@41082
   318
notation (xsymbols)
haftmann@41082
   319
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   320
haftmann@41082
   321
lemma Inter_eq:
haftmann@41082
   322
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   323
proof (rule set_eqI)
haftmann@41082
   324
  fix x
haftmann@41082
   325
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082
   326
    by auto
haftmann@41082
   327
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   328
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   329
qed
haftmann@41082
   330
haftmann@43741
   331
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   332
  by (unfold Inter_eq) blast
haftmann@41082
   333
haftmann@43741
   334
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   335
  by (simp add: Inter_eq)
haftmann@41082
   336
haftmann@41082
   337
text {*
haftmann@41082
   338
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   339
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   340
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   341
*}
haftmann@41082
   342
haftmann@43741
   343
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   344
  by auto
haftmann@41082
   345
haftmann@43741
   346
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   347
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   348
    @{prop "X \<in> C"}. *}
haftmann@41082
   349
  by (unfold Inter_eq) blast
haftmann@41082
   350
haftmann@43741
   351
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   352
  by (fact Inf_lower)
haftmann@43740
   353
haftmann@43740
   354
lemma (in complete_lattice) Inf_less_eq:
haftmann@43740
   355
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43740
   356
    and "A \<noteq> {}"
haftmann@43753
   357
  shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43740
   358
proof -
haftmann@43740
   359
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
haftmann@43740
   360
  moreover with assms have "v \<sqsubseteq> u" by blast
haftmann@43740
   361
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43740
   362
qed
haftmann@41082
   363
haftmann@41082
   364
lemma Inter_subset:
haftmann@43755
   365
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   366
  by (fact Inf_less_eq)
haftmann@41082
   367
haftmann@43755
   368
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   369
  by (fact Inf_greatest)
haftmann@41082
   370
haftmann@41082
   371
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   372
  by (fact Inf_binary [symmetric])
haftmann@41082
   373
haftmann@41082
   374
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   375
  by (fact Inf_empty)
haftmann@41082
   376
haftmann@41082
   377
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   378
  by (fact Inf_UNIV)
haftmann@41082
   379
haftmann@41082
   380
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   381
  by (fact Inf_insert)
haftmann@41082
   382
haftmann@43741
   383
lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43741
   384
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43741
   385
haftmann@41082
   386
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43741
   387
  by (fact Inf_inter_less)
haftmann@43741
   388
haftmann@43741
   389
(*lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"*)
haftmann@41082
   390
haftmann@41082
   391
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@41082
   392
  by blast
haftmann@41082
   393
haftmann@41082
   394
lemma Inter_UNIV_conv [simp,no_atp]:
haftmann@43741
   395
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   396
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@41082
   397
  by blast+
haftmann@41082
   398
haftmann@43741
   399
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@41082
   400
  by blast
haftmann@41082
   401
haftmann@41082
   402
haftmann@41082
   403
subsection {* Intersections of families *}
haftmann@41082
   404
haftmann@41082
   405
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   406
  "INTER \<equiv> INFI"
haftmann@41082
   407
haftmann@41082
   408
syntax
haftmann@41082
   409
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   410
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   411
haftmann@41082
   412
syntax (xsymbols)
haftmann@41082
   413
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   414
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   415
haftmann@41082
   416
syntax (latex output)
haftmann@41082
   417
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   418
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   419
haftmann@41082
   420
translations
haftmann@41082
   421
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   422
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   423
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   424
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   425
haftmann@41082
   426
print_translation {*
wenzelm@42284
   427
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   428
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   429
haftmann@41082
   430
lemma INTER_eq_Inter_image:
haftmann@41082
   431
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@41082
   432
  by (fact INFI_def)
haftmann@41082
   433
  
haftmann@41082
   434
lemma Inter_def:
haftmann@41082
   435
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   436
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   437
haftmann@41082
   438
lemma INTER_def:
haftmann@41082
   439
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   440
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   441
haftmann@41082
   442
lemma Inter_image_eq [simp]:
haftmann@41082
   443
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@41082
   444
  by (rule sym) (fact INTER_eq_Inter_image)
haftmann@41082
   445
haftmann@41082
   446
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
haftmann@41082
   447
  by (unfold INTER_def) blast
haftmann@41082
   448
haftmann@41082
   449
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
haftmann@41082
   450
  by (unfold INTER_def) blast
haftmann@41082
   451
haftmann@41082
   452
lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
haftmann@41082
   453
  by auto
haftmann@41082
   454
haftmann@41082
   455
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
haftmann@41082
   456
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
haftmann@41082
   457
  by (unfold INTER_def) blast
haftmann@41082
   458
haftmann@41082
   459
lemma INT_cong [cong]:
haftmann@41082
   460
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
haftmann@41082
   461
  by (simp add: INTER_def)
haftmann@41082
   462
haftmann@41082
   463
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   464
  by blast
haftmann@41082
   465
haftmann@41082
   466
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   467
  by blast
haftmann@41082
   468
haftmann@41082
   469
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   470
  by (fact INF_leI)
haftmann@41082
   471
haftmann@41082
   472
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@41082
   473
  by (fact le_INFI)
haftmann@41082
   474
haftmann@41082
   475
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@41082
   476
  by blast
haftmann@41082
   477
haftmann@41082
   478
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@41082
   479
  by blast
haftmann@41082
   480
haftmann@41082
   481
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   482
  by (fact le_INF_iff)
haftmann@41082
   483
haftmann@41082
   484
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@41082
   485
  by blast
haftmann@41082
   486
haftmann@41082
   487
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@41082
   488
  by blast
haftmann@41082
   489
haftmann@41082
   490
lemma INT_insert_distrib:
haftmann@41082
   491
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@41082
   492
  by blast
haftmann@41082
   493
haftmann@41082
   494
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@41082
   495
  by auto
haftmann@41082
   496
haftmann@41082
   497
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   498
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@41082
   499
  by blast
haftmann@41082
   500
haftmann@41082
   501
lemma INTER_UNIV_conv[simp]:
haftmann@41082
   502
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   503
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   504
by blast+
haftmann@41082
   505
haftmann@41082
   506
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@41082
   507
  by (auto intro: bool_induct)
haftmann@41082
   508
haftmann@41082
   509
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   510
  by blast
haftmann@41082
   511
haftmann@41082
   512
lemma INT_anti_mono:
haftmann@41082
   513
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@41082
   514
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@41082
   515
  -- {* The last inclusion is POSITIVE! *}
haftmann@41082
   516
  by (blast dest: subsetD)
haftmann@41082
   517
haftmann@41082
   518
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@41082
   519
  by blast
haftmann@41082
   520
haftmann@41082
   521
haftmann@32139
   522
subsection {* Union *}
haftmann@32115
   523
haftmann@32587
   524
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   525
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   526
haftmann@32115
   527
notation (xsymbols)
haftmann@32115
   528
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   529
haftmann@32135
   530
lemma Union_eq:
haftmann@32135
   531
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   532
proof (rule set_eqI)
haftmann@32115
   533
  fix x
haftmann@32135
   534
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115
   535
    by auto
haftmann@32135
   536
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   537
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   538
qed
haftmann@32115
   539
blanchet@35828
   540
lemma Union_iff [simp, no_atp]:
haftmann@32115
   541
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   542
  by (unfold Union_eq) blast
haftmann@32115
   543
haftmann@32115
   544
lemma UnionI [intro]:
haftmann@32115
   545
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   546
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   547
    @{term A} may be flexible. *}
haftmann@32115
   548
  by auto
haftmann@32115
   549
haftmann@32115
   550
lemma UnionE [elim!]:
haftmann@32115
   551
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   552
  by auto
haftmann@32115
   553
haftmann@32135
   554
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
haftmann@32135
   555
  by (iprover intro: subsetI UnionI)
haftmann@32135
   556
haftmann@32135
   557
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
haftmann@32135
   558
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135
   559
haftmann@32135
   560
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   561
  by blast
haftmann@32135
   562
haftmann@32135
   563
lemma Union_empty [simp]: "Union({}) = {}"
haftmann@32135
   564
  by blast
haftmann@32135
   565
haftmann@32135
   566
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
haftmann@32135
   567
  by blast
haftmann@32135
   568
haftmann@32135
   569
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
haftmann@32135
   570
  by blast
haftmann@32135
   571
haftmann@32135
   572
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
haftmann@32135
   573
  by blast
haftmann@32135
   574
haftmann@32135
   575
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135
   576
  by blast
haftmann@32135
   577
blanchet@35828
   578
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   579
  by blast
haftmann@32135
   580
blanchet@35828
   581
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   582
  by blast
haftmann@32135
   583
haftmann@32135
   584
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135
   585
  by blast
haftmann@32135
   586
haftmann@32135
   587
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   588
  by blast
haftmann@32135
   589
haftmann@32135
   590
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   591
  by blast
haftmann@32135
   592
haftmann@32135
   593
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
haftmann@32135
   594
  by blast
haftmann@32135
   595
haftmann@32115
   596
haftmann@32139
   597
subsection {* Unions of families *}
haftmann@32077
   598
haftmann@32606
   599
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   600
  "UNION \<equiv> SUPR"
haftmann@32077
   601
haftmann@32077
   602
syntax
wenzelm@35115
   603
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   604
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   605
haftmann@32077
   606
syntax (xsymbols)
wenzelm@35115
   607
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   608
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   609
haftmann@32077
   610
syntax (latex output)
wenzelm@35115
   611
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   612
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   613
haftmann@32077
   614
translations
haftmann@32077
   615
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   616
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   617
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   618
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   619
haftmann@32077
   620
text {*
haftmann@32077
   621
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   622
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   623
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   624
  former does not make the index expression a subscript of the
haftmann@32077
   625
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   626
  subscripts in Proof General.
haftmann@32077
   627
*}
haftmann@32077
   628
wenzelm@35115
   629
print_translation {*
wenzelm@42284
   630
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   631
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   632
haftmann@32135
   633
lemma UNION_eq_Union_image:
haftmann@32135
   634
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32606
   635
  by (fact SUPR_def)
haftmann@32115
   636
haftmann@32115
   637
lemma Union_def:
haftmann@32117
   638
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   639
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   640
blanchet@35828
   641
lemma UNION_def [no_atp]:
haftmann@32135
   642
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   643
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   644
  
haftmann@32115
   645
lemma Union_image_eq [simp]:
haftmann@32115
   646
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   647
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   648
  
wenzelm@11979
   649
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   650
  by (unfold UNION_def) blast
wenzelm@11979
   651
wenzelm@11979
   652
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   653
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   654
    @{term b} may be flexible. *}
wenzelm@11979
   655
  by auto
wenzelm@11979
   656
wenzelm@11979
   657
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   658
  by (unfold UNION_def) blast
clasohm@923
   659
wenzelm@11979
   660
lemma UN_cong [cong]:
wenzelm@11979
   661
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   662
  by (simp add: UNION_def)
wenzelm@11979
   663
berghofe@29691
   664
lemma strong_UN_cong:
berghofe@29691
   665
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691
   666
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   667
haftmann@32077
   668
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077
   669
  by blast
haftmann@32077
   670
haftmann@32135
   671
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606
   672
  by (fact le_SUPI)
haftmann@32135
   673
haftmann@32135
   674
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   675
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   676
blanchet@35828
   677
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   678
  by blast
haftmann@32135
   679
haftmann@32135
   680
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   681
  by blast
haftmann@32135
   682
blanchet@35828
   683
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   684
  by blast
haftmann@32135
   685
haftmann@32135
   686
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   687
  by blast
haftmann@32135
   688
haftmann@32135
   689
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   690
  by blast
haftmann@32135
   691
haftmann@32135
   692
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   693
  by auto
haftmann@32135
   694
haftmann@32135
   695
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   696
  by blast
haftmann@32135
   697
haftmann@32135
   698
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)"
haftmann@32135
   699
  by blast
haftmann@32135
   700
haftmann@32135
   701
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)"
haftmann@32135
   702
  by blast
haftmann@32135
   703
haftmann@32135
   704
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   705
  by (fact SUP_le_iff)
haftmann@32135
   706
haftmann@32135
   707
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   708
  by blast
haftmann@32135
   709
haftmann@32135
   710
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   711
  by auto
haftmann@32135
   712
haftmann@32135
   713
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   714
  by blast
haftmann@32135
   715
haftmann@32135
   716
lemma UNION_empty_conv[simp]:
haftmann@32135
   717
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   718
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   719
by blast+
haftmann@32135
   720
blanchet@35828
   721
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   722
  by blast
haftmann@32135
   723
haftmann@32135
   724
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   725
  by blast
haftmann@32135
   726
haftmann@32135
   727
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   728
  by blast
haftmann@32135
   729
haftmann@32135
   730
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   731
  by (auto simp add: split_if_mem2)
haftmann@32135
   732
haftmann@32135
   733
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135
   734
  by (auto intro: bool_contrapos)
haftmann@32135
   735
haftmann@32135
   736
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   737
  by blast
haftmann@32135
   738
haftmann@32135
   739
lemma UN_mono:
haftmann@32135
   740
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   741
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   742
  by (blast dest: subsetD)
haftmann@32135
   743
haftmann@32135
   744
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135
   745
  by blast
haftmann@32135
   746
haftmann@32135
   747
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135
   748
  by blast
haftmann@32135
   749
haftmann@32135
   750
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135
   751
  -- {* NOT suitable for rewriting *}
haftmann@32135
   752
  by blast
haftmann@32135
   753
haftmann@32135
   754
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135
   755
by blast
haftmann@32135
   756
wenzelm@11979
   757
haftmann@32139
   758
subsection {* Distributive laws *}
wenzelm@12897
   759
wenzelm@12897
   760
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   761
  by blast
wenzelm@12897
   762
wenzelm@12897
   763
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   764
  by blast
wenzelm@12897
   765
wenzelm@12897
   766
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
   767
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   768
  -- {* Union of a family of unions *}
wenzelm@12897
   769
  by blast
wenzelm@12897
   770
wenzelm@12897
   771
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)"
wenzelm@12897
   772
  -- {* Equivalent version *}
wenzelm@12897
   773
  by blast
wenzelm@12897
   774
wenzelm@12897
   775
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   776
  by blast
wenzelm@12897
   777
wenzelm@12897
   778
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
   779
  by blast
wenzelm@12897
   780
wenzelm@12897
   781
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)"
wenzelm@12897
   782
  -- {* Equivalent version *}
wenzelm@12897
   783
  by blast
wenzelm@12897
   784
wenzelm@12897
   785
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   786
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   787
  by blast
wenzelm@12897
   788
wenzelm@12897
   789
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   790
  by blast
wenzelm@12897
   791
wenzelm@12897
   792
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)"
wenzelm@12897
   793
  by blast
wenzelm@12897
   794
wenzelm@12897
   795
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)"
wenzelm@12897
   796
  by blast
wenzelm@12897
   797
wenzelm@12897
   798
haftmann@32139
   799
subsection {* Complement *}
haftmann@32135
   800
haftmann@32135
   801
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   802
  by blast
wenzelm@12897
   803
haftmann@32135
   804
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   805
  by blast
wenzelm@12897
   806
wenzelm@12897
   807
haftmann@32139
   808
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   809
paulson@13860
   810
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   811
           and Intersections. *}
wenzelm@12897
   812
wenzelm@12897
   813
lemma UN_simps [simp]:
wenzelm@12897
   814
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
   815
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
   816
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
   817
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
   818
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
   819
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
   820
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
   821
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
   822
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
   823
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   824
  by auto
wenzelm@12897
   825
wenzelm@12897
   826
lemma INT_simps [simp]:
wenzelm@12897
   827
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
   828
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
   829
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
   830
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
   831
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
   832
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
   833
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
   834
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
   835
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
   836
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   837
  by auto
wenzelm@12897
   838
blanchet@35828
   839
lemma ball_simps [simp,no_atp]:
wenzelm@12897
   840
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
   841
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
   842
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
   843
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
   844
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
   845
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
   846
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
   847
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
   848
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
   849
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
   850
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
   851
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
   852
  by auto
wenzelm@12897
   853
blanchet@35828
   854
lemma bex_simps [simp,no_atp]:
wenzelm@12897
   855
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
   856
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
   857
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
   858
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
   859
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
   860
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
   861
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
   862
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
   863
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
   864
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
   865
  by auto
wenzelm@12897
   866
wenzelm@12897
   867
lemma ball_conj_distrib:
wenzelm@12897
   868
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
   869
  by blast
wenzelm@12897
   870
wenzelm@12897
   871
lemma bex_disj_distrib:
wenzelm@12897
   872
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
   873
  by blast
wenzelm@12897
   874
wenzelm@12897
   875
paulson@13860
   876
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   877
paulson@13860
   878
lemma UN_extend_simps:
paulson@13860
   879
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
   880
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
   881
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
   882
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
   883
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
   884
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
   885
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
   886
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
   887
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
   888
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   889
  by auto
paulson@13860
   890
paulson@13860
   891
lemma INT_extend_simps:
paulson@13860
   892
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
   893
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
   894
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
   895
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
   896
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
   897
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
   898
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
   899
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
   900
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
   901
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   902
  by auto
paulson@13860
   903
paulson@13860
   904
haftmann@32135
   905
no_notation
haftmann@32135
   906
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   907
  less (infix "\<sqsubset>" 50) and
haftmann@41082
   908
  bot ("\<bottom>") and
haftmann@41082
   909
  top ("\<top>") and
haftmann@32135
   910
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   911
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   912
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
   913
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
   914
haftmann@41080
   915
no_syntax (xsymbols)
haftmann@41082
   916
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   917
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   918
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   919
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   920
haftmann@30596
   921
lemmas mem_simps =
haftmann@30596
   922
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   923
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   924
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   925
wenzelm@11979
   926
end