src/HOL/Complete_Lattice.thy
author haftmann
Wed Dec 08 15:05:46 2010 +0100 (2010-12-08)
changeset 41082 9ff94e7cc3b3
parent 41080 294956ff285b
child 41971 a54e8e95fe96
permissions -rw-r--r--
bot comes before top, inf before sup etc.
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
huffman@35629
    85
lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
huffman@35629
    86
  by (auto intro: Inf_greatest dest: Inf_lower)
huffman@35629
    87
haftmann@41082
    88
lemma Sup_le_iff: "Sup 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@38705
    92
  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
hoelzl@38705
    93
  shows "Inf A \<le> Inf B"
hoelzl@38705
    94
proof (rule Inf_greatest)
hoelzl@38705
    95
  fix b assume "b \<in> B"
hoelzl@38705
    96
  with assms obtain a where "a \<in> A" and "a \<le> b" by blast
hoelzl@38705
    97
  from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)
hoelzl@38705
    98
  with `a \<le> b` show "Inf A \<le> b" by auto
hoelzl@38705
    99
qed
hoelzl@38705
   100
haftmann@41082
   101
lemma Sup_mono:
haftmann@41082
   102
  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
haftmann@41082
   103
  shows "Sup A \<le> Sup B"
haftmann@41082
   104
proof (rule Sup_least)
haftmann@41082
   105
  fix a assume "a \<in> A"
haftmann@41082
   106
  with assms obtain b where "b \<in> B" and "a \<le> b" by blast
haftmann@41082
   107
  from `b \<in> B` have "b \<le> Sup B" by (rule Sup_upper)
haftmann@41082
   108
  with `a \<le> b` show "a \<le> Sup B" by auto
haftmann@41082
   109
qed
haftmann@32077
   110
haftmann@32077
   111
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@32117
   112
  "INFI A f = \<Sqinter> (f ` A)"
haftmann@32077
   113
haftmann@41082
   114
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@41082
   115
  "SUPR A f = \<Squnion> (f ` A)"
haftmann@41082
   116
haftmann@32077
   117
end
haftmann@32077
   118
haftmann@32077
   119
syntax
haftmann@41082
   120
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@41082
   121
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
haftmann@41080
   122
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@41080
   123
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
haftmann@41080
   124
haftmann@41080
   125
syntax (xsymbols)
haftmann@41082
   126
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   127
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   128
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   129
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   130
haftmann@32077
   131
translations
haftmann@41082
   132
  "INF x y. B"   == "INF x. INF y. B"
haftmann@41082
   133
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
haftmann@41082
   134
  "INF x. B"     == "INF x:CONST UNIV. B"
haftmann@41082
   135
  "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@32077
   136
  "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@32077
   137
  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
haftmann@32077
   138
  "SUP x. B"     == "SUP x:CONST UNIV. B"
haftmann@32077
   139
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@32077
   140
wenzelm@35115
   141
print_translation {*
haftmann@41082
   142
  [Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
haftmann@41082
   143
    Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
wenzelm@35115
   144
*} -- {* to avoid eta-contraction of body *}
wenzelm@11979
   145
haftmann@32077
   146
context complete_lattice
haftmann@32077
   147
begin
haftmann@32077
   148
haftmann@34007
   149
lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"
haftmann@32077
   150
  by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@32077
   151
haftmann@34007
   152
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"
haftmann@32077
   153
  by (auto simp add: SUPR_def intro: Sup_least)
haftmann@32077
   154
haftmann@34007
   155
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"
haftmann@32077
   156
  by (auto simp add: INFI_def intro: Inf_lower)
haftmann@32077
   157
haftmann@34007
   158
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
haftmann@32077
   159
  by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@32077
   160
huffman@35629
   161
lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
huffman@35629
   162
  unfolding SUPR_def by (auto simp add: Sup_le_iff)
huffman@35629
   163
huffman@35629
   164
lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
huffman@35629
   165
  unfolding INFI_def by (auto simp add: le_Inf_iff)
huffman@35629
   166
haftmann@32077
   167
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
haftmann@32077
   168
  by (auto intro: antisym INF_leI le_INFI)
haftmann@32077
   169
haftmann@41082
   170
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
haftmann@41082
   171
  by (auto intro: antisym SUP_leI le_SUPI)
hoelzl@38705
   172
hoelzl@38705
   173
lemma INF_mono:
hoelzl@38705
   174
  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
hoelzl@38705
   175
  by (force intro!: Inf_mono simp: INFI_def)
hoelzl@38705
   176
haftmann@41082
   177
lemma SUP_mono:
haftmann@41082
   178
  "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)"
haftmann@41082
   179
  by (force intro!: Sup_mono simp: SUPR_def)
hoelzl@40872
   180
hoelzl@40872
   181
lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
hoelzl@40872
   182
  by (intro INF_mono) auto
hoelzl@40872
   183
haftmann@41082
   184
lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"
haftmann@41082
   185
  by (intro SUP_mono) auto
hoelzl@40872
   186
hoelzl@40872
   187
lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
hoelzl@40872
   188
  by (iprover intro: INF_leI le_INFI order_trans antisym)
hoelzl@40872
   189
haftmann@41082
   190
lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
haftmann@41082
   191
  by (iprover intro: SUP_leI le_SUPI order_trans antisym)
haftmann@41082
   192
haftmann@32077
   193
end
haftmann@32077
   194
haftmann@41082
   195
lemma Inf_less_iff:
haftmann@41082
   196
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@41082
   197
  shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
haftmann@41082
   198
  unfolding not_le[symmetric] le_Inf_iff by auto
haftmann@41082
   199
hoelzl@38705
   200
lemma less_Sup_iff:
hoelzl@38705
   201
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
hoelzl@38705
   202
  shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
hoelzl@38705
   203
  unfolding not_le[symmetric] Sup_le_iff by auto
hoelzl@38705
   204
haftmann@41082
   205
lemma INF_less_iff:
haftmann@41082
   206
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@41082
   207
  shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
haftmann@41082
   208
  unfolding INFI_def Inf_less_iff by auto
haftmann@32077
   209
hoelzl@40872
   210
lemma less_SUP_iff:
hoelzl@40872
   211
  fixes a :: "'a::{complete_lattice,linorder}"
hoelzl@40872
   212
  shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
hoelzl@40872
   213
  unfolding SUPR_def less_Sup_iff by auto
hoelzl@40872
   214
haftmann@32139
   215
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   216
haftmann@32077
   217
instantiation bool :: complete_lattice
haftmann@32077
   218
begin
haftmann@32077
   219
haftmann@32077
   220
definition
haftmann@41080
   221
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077
   222
haftmann@32077
   223
definition
haftmann@41080
   224
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077
   225
haftmann@32077
   226
instance proof
haftmann@32077
   227
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@32077
   228
haftmann@32077
   229
end
haftmann@32077
   230
haftmann@41080
   231
lemma INFI_bool_eq [simp]:
haftmann@32120
   232
  "INFI = Ball"
haftmann@32120
   233
proof (rule ext)+
haftmann@32120
   234
  fix A :: "'a set"
haftmann@32120
   235
  fix P :: "'a \<Rightarrow> bool"
haftmann@32120
   236
  show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"
haftmann@32120
   237
    by (auto simp add: Ball_def INFI_def Inf_bool_def)
haftmann@32120
   238
qed
haftmann@32120
   239
haftmann@41080
   240
lemma SUPR_bool_eq [simp]:
haftmann@32120
   241
  "SUPR = Bex"
haftmann@32120
   242
proof (rule ext)+
haftmann@32120
   243
  fix A :: "'a set"
haftmann@32120
   244
  fix P :: "'a \<Rightarrow> bool"
haftmann@32120
   245
  show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
haftmann@32120
   246
    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
haftmann@32120
   247
qed
haftmann@32120
   248
haftmann@32077
   249
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   250
begin
haftmann@32077
   251
haftmann@32077
   252
definition
haftmann@41080
   253
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   254
haftmann@41080
   255
lemma Inf_apply:
haftmann@41080
   256
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   257
  by (simp add: Inf_fun_def)
haftmann@32077
   258
haftmann@32077
   259
definition
haftmann@41080
   260
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   261
haftmann@41080
   262
lemma Sup_apply:
haftmann@41080
   263
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   264
  by (simp add: Sup_fun_def)
haftmann@32077
   265
haftmann@32077
   266
instance proof
haftmann@41080
   267
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   268
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   269
haftmann@32077
   270
end
haftmann@32077
   271
haftmann@41080
   272
lemma INFI_apply:
haftmann@41080
   273
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@41080
   274
  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
hoelzl@38705
   275
haftmann@41080
   276
lemma SUPR_apply:
haftmann@41080
   277
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@41080
   278
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
haftmann@32077
   279
haftmann@32077
   280
haftmann@41082
   281
subsection {* Inter *}
haftmann@41082
   282
haftmann@41082
   283
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   284
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   285
  
haftmann@41082
   286
notation (xsymbols)
haftmann@41082
   287
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   288
haftmann@41082
   289
lemma Inter_eq:
haftmann@41082
   290
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   291
proof (rule set_eqI)
haftmann@41082
   292
  fix x
haftmann@41082
   293
  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
   294
    by auto
haftmann@41082
   295
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   296
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   297
qed
haftmann@41082
   298
haftmann@41082
   299
lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
haftmann@41082
   300
  by (unfold Inter_eq) blast
haftmann@41082
   301
haftmann@41082
   302
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
haftmann@41082
   303
  by (simp add: Inter_eq)
haftmann@41082
   304
haftmann@41082
   305
text {*
haftmann@41082
   306
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@41082
   307
  contains @{term A} as an element, but @{prop "A:X"} can hold when
haftmann@41082
   308
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   309
*}
haftmann@41082
   310
haftmann@41082
   311
lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
haftmann@41082
   312
  by auto
haftmann@41082
   313
haftmann@41082
   314
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
haftmann@41082
   315
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@41082
   316
    @{prop "X:C"}. *}
haftmann@41082
   317
  by (unfold Inter_eq) blast
haftmann@41082
   318
haftmann@41082
   319
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
haftmann@41082
   320
  by blast
haftmann@41082
   321
haftmann@41082
   322
lemma Inter_subset:
haftmann@41082
   323
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
haftmann@41082
   324
  by blast
haftmann@41082
   325
haftmann@41082
   326
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
haftmann@41082
   327
  by (iprover intro: InterI subsetI dest: subsetD)
haftmann@41082
   328
haftmann@41082
   329
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@41082
   330
  by blast
haftmann@41082
   331
haftmann@41082
   332
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   333
  by (fact Inf_empty)
haftmann@41082
   334
haftmann@41082
   335
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@41082
   336
  by blast
haftmann@41082
   337
haftmann@41082
   338
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@41082
   339
  by blast
haftmann@41082
   340
haftmann@41082
   341
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@41082
   342
  by blast
haftmann@41082
   343
haftmann@41082
   344
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@41082
   345
  by blast
haftmann@41082
   346
haftmann@41082
   347
lemma Inter_UNIV_conv [simp,no_atp]:
haftmann@41082
   348
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
haftmann@41082
   349
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
haftmann@41082
   350
  by blast+
haftmann@41082
   351
haftmann@41082
   352
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
haftmann@41082
   353
  by blast
haftmann@41082
   354
haftmann@41082
   355
haftmann@41082
   356
subsection {* Intersections of families *}
haftmann@41082
   357
haftmann@41082
   358
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   359
  "INTER \<equiv> INFI"
haftmann@41082
   360
haftmann@41082
   361
syntax
haftmann@41082
   362
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   363
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   364
haftmann@41082
   365
syntax (xsymbols)
haftmann@41082
   366
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   367
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   368
haftmann@41082
   369
syntax (latex output)
haftmann@41082
   370
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   371
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   372
haftmann@41082
   373
translations
haftmann@41082
   374
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   375
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   376
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   377
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   378
haftmann@41082
   379
print_translation {*
haftmann@41082
   380
  [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   381
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   382
haftmann@41082
   383
lemma INTER_eq_Inter_image:
haftmann@41082
   384
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@41082
   385
  by (fact INFI_def)
haftmann@41082
   386
  
haftmann@41082
   387
lemma Inter_def:
haftmann@41082
   388
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   389
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   390
haftmann@41082
   391
lemma INTER_def:
haftmann@41082
   392
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   393
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   394
haftmann@41082
   395
lemma Inter_image_eq [simp]:
haftmann@41082
   396
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@41082
   397
  by (rule sym) (fact INTER_eq_Inter_image)
haftmann@41082
   398
haftmann@41082
   399
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
haftmann@41082
   400
  by (unfold INTER_def) blast
haftmann@41082
   401
haftmann@41082
   402
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
haftmann@41082
   403
  by (unfold INTER_def) blast
haftmann@41082
   404
haftmann@41082
   405
lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
haftmann@41082
   406
  by auto
haftmann@41082
   407
haftmann@41082
   408
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
haftmann@41082
   409
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
haftmann@41082
   410
  by (unfold INTER_def) blast
haftmann@41082
   411
haftmann@41082
   412
lemma INT_cong [cong]:
haftmann@41082
   413
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
haftmann@41082
   414
  by (simp add: INTER_def)
haftmann@41082
   415
haftmann@41082
   416
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   417
  by blast
haftmann@41082
   418
haftmann@41082
   419
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   420
  by blast
haftmann@41082
   421
haftmann@41082
   422
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   423
  by (fact INF_leI)
haftmann@41082
   424
haftmann@41082
   425
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@41082
   426
  by (fact le_INFI)
haftmann@41082
   427
haftmann@41082
   428
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@41082
   429
  by blast
haftmann@41082
   430
haftmann@41082
   431
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@41082
   432
  by blast
haftmann@41082
   433
haftmann@41082
   434
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   435
  by (fact le_INF_iff)
haftmann@41082
   436
haftmann@41082
   437
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@41082
   438
  by blast
haftmann@41082
   439
haftmann@41082
   440
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
   441
  by blast
haftmann@41082
   442
haftmann@41082
   443
lemma INT_insert_distrib:
haftmann@41082
   444
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@41082
   445
  by blast
haftmann@41082
   446
haftmann@41082
   447
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@41082
   448
  by auto
haftmann@41082
   449
haftmann@41082
   450
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   451
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@41082
   452
  by blast
haftmann@41082
   453
haftmann@41082
   454
lemma INTER_UNIV_conv[simp]:
haftmann@41082
   455
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   456
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   457
by blast+
haftmann@41082
   458
haftmann@41082
   459
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@41082
   460
  by (auto intro: bool_induct)
haftmann@41082
   461
haftmann@41082
   462
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   463
  by blast
haftmann@41082
   464
haftmann@41082
   465
lemma INT_anti_mono:
haftmann@41082
   466
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@41082
   467
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@41082
   468
  -- {* The last inclusion is POSITIVE! *}
haftmann@41082
   469
  by (blast dest: subsetD)
haftmann@41082
   470
haftmann@41082
   471
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@41082
   472
  by blast
haftmann@41082
   473
haftmann@41082
   474
haftmann@32139
   475
subsection {* Union *}
haftmann@32115
   476
haftmann@32587
   477
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   478
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   479
haftmann@32115
   480
notation (xsymbols)
haftmann@32115
   481
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   482
haftmann@32135
   483
lemma Union_eq:
haftmann@32135
   484
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   485
proof (rule set_eqI)
haftmann@32115
   486
  fix x
haftmann@32135
   487
  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
   488
    by auto
haftmann@32135
   489
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   490
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   491
qed
haftmann@32115
   492
blanchet@35828
   493
lemma Union_iff [simp, no_atp]:
haftmann@32115
   494
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   495
  by (unfold Union_eq) blast
haftmann@32115
   496
haftmann@32115
   497
lemma UnionI [intro]:
haftmann@32115
   498
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   499
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   500
    @{term A} may be flexible. *}
haftmann@32115
   501
  by auto
haftmann@32115
   502
haftmann@32115
   503
lemma UnionE [elim!]:
haftmann@32115
   504
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   505
  by auto
haftmann@32115
   506
haftmann@32135
   507
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
haftmann@32135
   508
  by (iprover intro: subsetI UnionI)
haftmann@32135
   509
haftmann@32135
   510
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
haftmann@32135
   511
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135
   512
haftmann@32135
   513
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   514
  by blast
haftmann@32135
   515
haftmann@32135
   516
lemma Union_empty [simp]: "Union({}) = {}"
haftmann@32135
   517
  by blast
haftmann@32135
   518
haftmann@32135
   519
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
haftmann@32135
   520
  by blast
haftmann@32135
   521
haftmann@32135
   522
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
haftmann@32135
   523
  by blast
haftmann@32135
   524
haftmann@32135
   525
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
haftmann@32135
   526
  by blast
haftmann@32135
   527
haftmann@32135
   528
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135
   529
  by blast
haftmann@32135
   530
blanchet@35828
   531
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   532
  by blast
haftmann@32135
   533
blanchet@35828
   534
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   535
  by blast
haftmann@32135
   536
haftmann@32135
   537
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135
   538
  by blast
haftmann@32135
   539
haftmann@32135
   540
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   541
  by blast
haftmann@32135
   542
haftmann@32135
   543
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   544
  by blast
haftmann@32135
   545
haftmann@32135
   546
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
haftmann@32135
   547
  by blast
haftmann@32135
   548
haftmann@32115
   549
haftmann@32139
   550
subsection {* Unions of families *}
haftmann@32077
   551
haftmann@32606
   552
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   553
  "UNION \<equiv> SUPR"
haftmann@32077
   554
haftmann@32077
   555
syntax
wenzelm@35115
   556
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   557
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   558
haftmann@32077
   559
syntax (xsymbols)
wenzelm@35115
   560
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   561
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   562
haftmann@32077
   563
syntax (latex output)
wenzelm@35115
   564
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   565
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   566
haftmann@32077
   567
translations
haftmann@32077
   568
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   569
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   570
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   571
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   572
haftmann@32077
   573
text {*
haftmann@32077
   574
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   575
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   576
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   577
  former does not make the index expression a subscript of the
haftmann@32077
   578
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   579
  subscripts in Proof General.
haftmann@32077
   580
*}
haftmann@32077
   581
wenzelm@35115
   582
print_translation {*
wenzelm@35115
   583
  [Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   584
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   585
haftmann@32135
   586
lemma UNION_eq_Union_image:
haftmann@32135
   587
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32606
   588
  by (fact SUPR_def)
haftmann@32115
   589
haftmann@32115
   590
lemma Union_def:
haftmann@32117
   591
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   592
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   593
blanchet@35828
   594
lemma UNION_def [no_atp]:
haftmann@32135
   595
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   596
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   597
  
haftmann@32115
   598
lemma Union_image_eq [simp]:
haftmann@32115
   599
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   600
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   601
  
wenzelm@11979
   602
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   603
  by (unfold UNION_def) blast
wenzelm@11979
   604
wenzelm@11979
   605
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   606
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   607
    @{term b} may be flexible. *}
wenzelm@11979
   608
  by auto
wenzelm@11979
   609
wenzelm@11979
   610
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   611
  by (unfold UNION_def) blast
clasohm@923
   612
wenzelm@11979
   613
lemma UN_cong [cong]:
wenzelm@11979
   614
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   615
  by (simp add: UNION_def)
wenzelm@11979
   616
berghofe@29691
   617
lemma strong_UN_cong:
berghofe@29691
   618
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691
   619
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   620
haftmann@32077
   621
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077
   622
  by blast
haftmann@32077
   623
haftmann@32135
   624
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606
   625
  by (fact le_SUPI)
haftmann@32135
   626
haftmann@32135
   627
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   628
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   629
blanchet@35828
   630
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   631
  by blast
haftmann@32135
   632
haftmann@32135
   633
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
   634
  by blast
haftmann@32135
   635
blanchet@35828
   636
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   637
  by blast
haftmann@32135
   638
haftmann@32135
   639
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   640
  by blast
haftmann@32135
   641
haftmann@32135
   642
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   643
  by blast
haftmann@32135
   644
haftmann@32135
   645
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   646
  by auto
haftmann@32135
   647
haftmann@32135
   648
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   649
  by blast
haftmann@32135
   650
haftmann@32135
   651
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
   652
  by blast
haftmann@32135
   653
haftmann@32135
   654
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
   655
  by blast
haftmann@32135
   656
haftmann@32135
   657
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   658
  by (fact SUP_le_iff)
haftmann@32135
   659
haftmann@32135
   660
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   661
  by blast
haftmann@32135
   662
haftmann@32135
   663
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   664
  by auto
haftmann@32135
   665
haftmann@32135
   666
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   667
  by blast
haftmann@32135
   668
haftmann@32135
   669
lemma UNION_empty_conv[simp]:
haftmann@32135
   670
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   671
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   672
by blast+
haftmann@32135
   673
blanchet@35828
   674
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   675
  by blast
haftmann@32135
   676
haftmann@32135
   677
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   678
  by blast
haftmann@32135
   679
haftmann@32135
   680
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   681
  by blast
haftmann@32135
   682
haftmann@32135
   683
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   684
  by (auto simp add: split_if_mem2)
haftmann@32135
   685
haftmann@32135
   686
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135
   687
  by (auto intro: bool_contrapos)
haftmann@32135
   688
haftmann@32135
   689
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   690
  by blast
haftmann@32135
   691
haftmann@32135
   692
lemma UN_mono:
haftmann@32135
   693
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   694
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   695
  by (blast dest: subsetD)
haftmann@32135
   696
haftmann@32135
   697
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135
   698
  by blast
haftmann@32135
   699
haftmann@32135
   700
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135
   701
  by blast
haftmann@32135
   702
haftmann@32135
   703
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135
   704
  -- {* NOT suitable for rewriting *}
haftmann@32135
   705
  by blast
haftmann@32135
   706
haftmann@32135
   707
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135
   708
by blast
haftmann@32135
   709
wenzelm@11979
   710
haftmann@32139
   711
subsection {* Distributive laws *}
wenzelm@12897
   712
wenzelm@12897
   713
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   714
  by blast
wenzelm@12897
   715
wenzelm@12897
   716
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   717
  by blast
wenzelm@12897
   718
wenzelm@12897
   719
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
   720
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   721
  -- {* Union of a family of unions *}
wenzelm@12897
   722
  by blast
wenzelm@12897
   723
wenzelm@12897
   724
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
   725
  -- {* Equivalent version *}
wenzelm@12897
   726
  by blast
wenzelm@12897
   727
wenzelm@12897
   728
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   729
  by blast
wenzelm@12897
   730
wenzelm@12897
   731
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
   732
  by blast
wenzelm@12897
   733
wenzelm@12897
   734
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
   735
  -- {* Equivalent version *}
wenzelm@12897
   736
  by blast
wenzelm@12897
   737
wenzelm@12897
   738
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   739
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   740
  by blast
wenzelm@12897
   741
wenzelm@12897
   742
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   743
  by blast
wenzelm@12897
   744
wenzelm@12897
   745
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
   746
  by blast
wenzelm@12897
   747
wenzelm@12897
   748
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
   749
  by blast
wenzelm@12897
   750
wenzelm@12897
   751
haftmann@32139
   752
subsection {* Complement *}
haftmann@32135
   753
haftmann@32135
   754
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   755
  by blast
wenzelm@12897
   756
haftmann@32135
   757
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   758
  by blast
wenzelm@12897
   759
wenzelm@12897
   760
haftmann@32139
   761
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   762
paulson@13860
   763
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   764
           and Intersections. *}
wenzelm@12897
   765
wenzelm@12897
   766
lemma UN_simps [simp]:
wenzelm@12897
   767
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
   768
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
   769
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
   770
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
   771
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
   772
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
   773
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
   774
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
   775
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
   776
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   777
  by auto
wenzelm@12897
   778
wenzelm@12897
   779
lemma INT_simps [simp]:
wenzelm@12897
   780
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
   781
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
   782
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
   783
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
   784
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
   785
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
   786
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
   787
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
   788
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
   789
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   790
  by auto
wenzelm@12897
   791
blanchet@35828
   792
lemma ball_simps [simp,no_atp]:
wenzelm@12897
   793
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
   794
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
   795
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
   796
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
   797
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
   798
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
   799
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
   800
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
   801
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
   802
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
   803
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
   804
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
   805
  by auto
wenzelm@12897
   806
blanchet@35828
   807
lemma bex_simps [simp,no_atp]:
wenzelm@12897
   808
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
   809
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
   810
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
   811
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
   812
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
   813
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
   814
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
   815
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
   816
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
   817
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
   818
  by auto
wenzelm@12897
   819
wenzelm@12897
   820
lemma ball_conj_distrib:
wenzelm@12897
   821
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
   822
  by blast
wenzelm@12897
   823
wenzelm@12897
   824
lemma bex_disj_distrib:
wenzelm@12897
   825
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
   826
  by blast
wenzelm@12897
   827
wenzelm@12897
   828
paulson@13860
   829
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   830
paulson@13860
   831
lemma UN_extend_simps:
paulson@13860
   832
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
   833
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
   834
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
   835
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
   836
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
   837
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
   838
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
   839
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
   840
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
   841
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   842
  by auto
paulson@13860
   843
paulson@13860
   844
lemma INT_extend_simps:
paulson@13860
   845
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
   846
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
   847
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
   848
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
   849
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
   850
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
   851
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
   852
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
   853
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
   854
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   855
  by auto
paulson@13860
   856
paulson@13860
   857
haftmann@32135
   858
no_notation
haftmann@32135
   859
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   860
  less (infix "\<sqsubset>" 50) and
haftmann@41082
   861
  bot ("\<bottom>") and
haftmann@41082
   862
  top ("\<top>") and
haftmann@32135
   863
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   864
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   865
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
   866
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
   867
haftmann@41080
   868
no_syntax (xsymbols)
haftmann@41082
   869
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   870
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   871
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   872
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   873
haftmann@30596
   874
lemmas mem_simps =
haftmann@30596
   875
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   876
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   877
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   878
wenzelm@11979
   879
end