src/HOL/Complete_Lattice.thy
 author haftmann Sat Jul 16 22:28:35 2011 +0200 (2011-07-16) changeset 43854 f1d23df1adde parent 43853 020ddc6a9508 child 43865 db18f4d0cc7d permissions -rw-r--r--
generalized some lemmas
     1 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)

     2

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

     4

     5 theory Complete_Lattice

     6 imports Set

     7 begin

     8

     9 notation

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

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

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

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

    14   top ("\<top>") and

    15   bot ("\<bottom>")

    16

    17

    18 subsection {* Syntactic infimum and supremum operations *}

    19

    20 class Inf =

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

    22

    23 class Sup =

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

    25

    26 subsection {* Abstract complete lattices *}

    27

    28 class complete_lattice = bounded_lattice + Inf + Sup +

    29   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"

    30      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"

    31   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"

    32      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"

    33 begin

    34

    35 lemma dual_complete_lattice:

    36   "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"

    37   by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)

    38     (unfold_locales, (fact bot_least top_greatest

    39         Sup_upper Sup_least Inf_lower Inf_greatest)+)

    40

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

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

    43

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

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

    46

    47 lemma Inf_empty [simp]:

    48   "\<Sqinter>{} = \<top>"

    49   by (auto intro: antisym Inf_greatest)

    50

    51 lemma Sup_empty [simp]:

    52   "\<Squnion>{} = \<bottom>"

    53   by (auto intro: antisym Sup_least)

    54

    55 lemma Inf_UNIV [simp]:

    56   "\<Sqinter>UNIV = \<bottom>"

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

    58

    59 lemma Sup_UNIV [simp]:

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

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

    62

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

    64   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)

    65

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

    67   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)

    68

    69 lemma Inf_singleton [simp]:

    70   "\<Sqinter>{a} = a"

    71   by (auto intro: antisym Inf_lower Inf_greatest)

    72

    73 lemma Sup_singleton [simp]:

    74   "\<Squnion>{a} = a"

    75   by (auto intro: antisym Sup_upper Sup_least)

    76

    77 lemma Inf_binary:

    78   "\<Sqinter>{a, b} = a \<sqinter> b"

    79   by (simp add: Inf_insert)

    80

    81 lemma Sup_binary:

    82   "\<Squnion>{a, b} = a \<squnion> b"

    83   by (simp add: Sup_insert)

    84

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

    86   by (auto intro: Inf_greatest dest: Inf_lower)

    87

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

    89   by (auto intro: Sup_least dest: Sup_upper)

    90

    91 lemma Inf_mono:

    92   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"

    93   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"

    94 proof (rule Inf_greatest)

    95   fix b assume "b \<in> B"

    96   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast

    97   from a \<in> A have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)

    98   with a \<sqsubseteq> b show "\<Sqinter>A \<sqsubseteq> b" by auto

    99 qed

   100

   101 lemma Sup_mono:

   102   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"

   103   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"

   104 proof (rule Sup_least)

   105   fix a assume "a \<in> A"

   106   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast

   107   from b \<in> B have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)

   108   with a \<sqsubseteq> b show "a \<sqsubseteq> \<Squnion>B" by auto

   109 qed

   110

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

   112   using Sup_upper[of u A] by auto

   113

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

   115   using Inf_lower[of u A] by auto

   116

   117 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where

   118   "INFI A f = \<Sqinter> (f  A)"

   119

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

   121   "SUPR A f = \<Squnion> (f  A)"

   122

   123 end

   124

   125 syntax

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

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

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

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

   130

   131 syntax (xsymbols)

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

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

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

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

   136

   137 translations

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

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

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

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

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

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

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

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

   146

   147 print_translation {*

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

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

   150 *} -- {* to avoid eta-contraction of body *}

   151

   152 context complete_lattice

   153 begin

   154

   155 lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"

   156   by (simp add: SUPR_def cong: image_cong)

   157

   158 lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"

   159   by (simp add: INFI_def cong: image_cong)

   160

   161 lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"

   162   by (auto simp add: SUPR_def intro: Sup_upper)

   163

   164 lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"

   165   using le_SUPI[of i A M] by auto

   166

   167 lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"

   168   by (auto simp add: SUPR_def intro: Sup_least)

   169

   170 lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"

   171   by (auto simp add: INFI_def intro: Inf_lower)

   172

   173 lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"

   174   using INF_leI[of i A M] by auto

   175

   176 lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"

   177   by (auto simp add: INFI_def intro: Inf_greatest)

   178

   179 lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"

   180   unfolding SUPR_def by (auto simp add: Sup_le_iff)

   181

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

   183   unfolding INFI_def by (auto simp add: le_Inf_iff)

   184

   185 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"

   186   by (auto intro: antisym INF_leI le_INFI)

   187

   188 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"

   189   by (auto intro: antisym SUP_leI le_SUPI)

   190

   191 lemma INF_mono:

   192   "(\<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)"

   193   by (force intro!: Inf_mono simp: INFI_def)

   194

   195 lemma SUP_mono:

   196   "(\<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)"

   197   by (force intro!: Sup_mono simp: SUPR_def)

   198

   199 lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"

   200   by (intro INF_mono) auto

   201

   202 lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"

   203   by (intro SUP_mono) auto

   204

   205 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)"

   206   by (iprover intro: INF_leI le_INFI order_trans antisym)

   207

   208 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)"

   209   by (iprover intro: SUP_leI le_SUPI order_trans antisym)

   210

   211 lemma INFI_insert: "(\<Sqinter>x\<in>insert a A. B x) = B a \<sqinter> INFI A B"

   212   by (simp add: INFI_def Inf_insert)

   213

   214 lemma SUPR_insert: "(\<Squnion>x\<in>insert a A. B x) = B a \<squnion> SUPR A B"

   215   by (simp add: SUPR_def Sup_insert)

   216

   217 end

   218

   219 lemma Inf_less_iff:

   220   fixes a :: "'a\<Colon>{complete_lattice,linorder}"

   221   shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"

   222   unfolding not_le [symmetric] le_Inf_iff by auto

   223

   224 lemma less_Sup_iff:

   225   fixes a :: "'a\<Colon>{complete_lattice,linorder}"

   226   shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"

   227   unfolding not_le [symmetric] Sup_le_iff by auto

   228

   229 lemma INF_less_iff:

   230   fixes a :: "'a::{complete_lattice,linorder}"

   231   shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"

   232   unfolding INFI_def Inf_less_iff by auto

   233

   234 lemma less_SUP_iff:

   235   fixes a :: "'a::{complete_lattice,linorder}"

   236   shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"

   237   unfolding SUPR_def less_Sup_iff by auto

   238

   239 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}

   240

   241 instantiation bool :: complete_lattice

   242 begin

   243

   244 definition

   245   "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"

   246

   247 definition

   248   "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"

   249

   250 instance proof

   251 qed (auto simp add: Inf_bool_def Sup_bool_def)

   252

   253 end

   254

   255 lemma INFI_bool_eq [simp]:

   256   "INFI = Ball"

   257 proof (rule ext)+

   258   fix A :: "'a set"

   259   fix P :: "'a \<Rightarrow> bool"

   260   show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"

   261     by (auto simp add: Ball_def INFI_def Inf_bool_def)

   262 qed

   263

   264 lemma SUPR_bool_eq [simp]:

   265   "SUPR = Bex"

   266 proof (rule ext)+

   267   fix A :: "'a set"

   268   fix P :: "'a \<Rightarrow> bool"

   269   show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"

   270     by (auto simp add: Bex_def SUPR_def Sup_bool_def)

   271 qed

   272

   273 instantiation "fun" :: (type, complete_lattice) complete_lattice

   274 begin

   275

   276 definition

   277   "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"

   278

   279 lemma Inf_apply:

   280   "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"

   281   by (simp add: Inf_fun_def)

   282

   283 definition

   284   "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"

   285

   286 lemma Sup_apply:

   287   "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"

   288   by (simp add: Sup_fun_def)

   289

   290 instance proof

   291 qed (auto simp add: le_fun_def Inf_apply Sup_apply

   292   intro: Inf_lower Sup_upper Inf_greatest Sup_least)

   293

   294 end

   295

   296 lemma INFI_apply:

   297   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"

   298   by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)

   299

   300 lemma SUPR_apply:

   301   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"

   302   by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)

   303

   304

   305 subsection {* Inter *}

   306

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

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

   309

   310 notation (xsymbols)

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

   312

   313 lemma Inter_eq:

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

   315 proof (rule set_eqI)

   316   fix x

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

   318     by auto

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

   320     by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)

   321 qed

   322

   323 lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"

   324   by (unfold Inter_eq) blast

   325

   326 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"

   327   by (simp add: Inter_eq)

   328

   329 text {*

   330   \medskip A destruct'' rule -- every @{term X} in @{term C}

   331   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when

   332   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.

   333 *}

   334

   335 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"

   336   by auto

   337

   338 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"

   339   -- {* Classical'' elimination rule -- does not require proving

   340     @{prop "X \<in> C"}. *}

   341   by (unfold Inter_eq) blast

   342

   343 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"

   344   by (fact Inf_lower)

   345

   346 lemma (in complete_lattice) Inf_less_eq:

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

   348     and "A \<noteq> {}"

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

   350 proof -

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

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

   353   ultimately show ?thesis by (rule Inf_lower2)

   354 qed

   355

   356 lemma Inter_subset:

   357   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"

   358   by (fact Inf_less_eq)

   359

   360 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"

   361   by (fact Inf_greatest)

   362

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

   364   by (fact Inf_binary [symmetric])

   365

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

   367   by (fact Inf_empty)

   368

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

   370   by (fact Inf_UNIV)

   371

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

   373   by (fact Inf_insert)

   374

   375 lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"

   376   by (auto intro: Inf_greatest Inf_lower)

   377

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

   379   by (fact Inf_inter_less)

   380

   381 lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"

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

   383

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

   385   by (fact Inf_union_distrib)

   386

   387 lemma (in complete_lattice) Inf_top_conv [no_atp]:

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

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

   390 proof -

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

   392   proof

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

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

   395     then show "\<Sqinter>A = \<top>" by auto

   396   next

   397     assume "\<Sqinter>A = \<top>"

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

   399     proof (rule ccontr)

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

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

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

   403       with \<Sqinter>A = \<top> x \<noteq> \<top> show False by (simp add: Inf_insert)

   404     qed

   405   qed

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

   407 qed

   408

   409 lemma Inter_UNIV_conv [simp,no_atp]:

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

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

   412   by (fact Inf_top_conv)+

   413

   414 lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"

   415   by (auto intro: Inf_greatest Inf_lower)

   416

   417 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"

   418   by (fact Inf_anti_mono)

   419

   420

   421 subsection {* Intersections of families *}

   422

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

   424   "INTER \<equiv> INFI"

   425

   426 syntax

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

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

   429

   430 syntax (xsymbols)

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

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

   433

   434 syntax (latex output)

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

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

   437

   438 translations

   439   "INT x y. B"  == "INT x. INT y. B"

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

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

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

   443

   444 print_translation {*

   445   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]

   446 *} -- {* to avoid eta-contraction of body *}

   447

   448 lemma INTER_eq_Inter_image:

   449   "(\<Inter>x\<in>A. B x) = \<Inter>(BA)"

   450   by (fact INFI_def)

   451

   452 lemma Inter_def:

   453   "\<Inter>S = (\<Inter>x\<in>S. x)"

   454   by (simp add: INTER_eq_Inter_image image_def)

   455

   456 lemma INTER_def:

   457   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"

   458   by (auto simp add: INTER_eq_Inter_image Inter_eq)

   459

   460 lemma Inter_image_eq [simp]:

   461   "\<Inter>(BA) = (\<Inter>x\<in>A. B x)"

   462   by (rule sym) (fact INFI_def)

   463

   464 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"

   465   by (unfold INTER_def) blast

   466

   467 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"

   468   by (unfold INTER_def) blast

   469

   470 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"

   471   by auto

   472

   473 lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"

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

   475   by (unfold INTER_def) blast

   476

   477 lemma (in complete_lattice) INFI_cong:

   478   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"

   479   by (simp add: INFI_def image_def)

   480

   481 lemma INT_cong [cong]:

   482   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"

   483   by (fact INFI_cong)

   484

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

   486   by blast

   487

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

   489   by blast

   490

   491 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"

   492   by (fact INF_leI)

   493

   494 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"

   495   by (fact le_INFI)

   496

   497 lemma (in complete_lattice) INFI_empty:

   498   "(\<Sqinter>x\<in>{}. B x) = \<top>"

   499   by (simp add: INFI_def)

   500

   501 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"

   502   by (fact INFI_empty)

   503

   504 lemma (in complete_lattice) INFI_absorb:

   505   assumes "k \<in> I"

   506   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"

   507 proof -

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

   509   then show ?thesis by (simp add: INFI_insert)

   510 qed

   511

   512 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"

   513   by (fact INFI_absorb)

   514

   515 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"

   516   by (fact le_INF_iff)

   517

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

   519   by (fact INFI_insert)

   520

   521 -- {* continue generalization from here *}

   522

   523 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"

   524   by blast

   525

   526 lemma INT_insert_distrib:

   527     "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"

   528   by blast

   529

   530 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"

   531   by auto

   532

   533 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"

   534   -- {* Look: it has an \emph{existential} quantifier *}

   535   by blast

   536

   537 lemma INTER_UNIV_conv [simp]:

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

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

   540   by blast+

   541

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

   543   by (auto intro: bool_induct)

   544

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

   546   by blast

   547

   548 lemma INT_anti_mono:

   549   "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>

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

   551   -- {* The last inclusion is POSITIVE! *}

   552   by (blast dest: subsetD)

   553

   554 lemma vimage_INT: "f - (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f - B x)"

   555   by blast

   556

   557

   558 subsection {* Union *}

   559

   560 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where

   561   "Union S \<equiv> \<Squnion>S"

   562

   563 notation (xsymbols)

   564   Union  ("\<Union>_" [90] 90)

   565

   566 lemma Union_eq:

   567   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"

   568 proof (rule set_eqI)

   569   fix x

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

   571     by auto

   572   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"

   573     by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)

   574 qed

   575

   576 lemma Union_iff [simp, no_atp]:

   577   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"

   578   by (unfold Union_eq) blast

   579

   580 lemma UnionI [intro]:

   581   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"

   582   -- {* The order of the premises presupposes that @{term C} is rigid;

   583     @{term A} may be flexible. *}

   584   by auto

   585

   586 lemma UnionE [elim!]:

   587   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"

   588   by auto

   589

   590 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"

   591   by (iprover intro: subsetI UnionI)

   592

   593 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"

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

   595

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

   597   by blast

   598

   599 lemma Union_empty [simp]: "\<Union>{} = {}"

   600   by blast

   601

   602 lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"

   603   by blast

   604

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

   606   by blast

   607

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

   609   by blast

   610

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

   612   by blast

   613

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

   615   by blast

   616

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

   618   by blast

   619

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

   621   by blast

   622

   623 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"

   624   by blast

   625

   626 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"

   627   by blast

   628

   629 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"

   630   by blast

   631

   632

   633 subsection {* Unions of families *}

   634

   635 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   636   "UNION \<equiv> SUPR"

   637

   638 syntax

   639   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)

   640   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)

   641

   642 syntax (xsymbols)

   643   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)

   644   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)

   645

   646 syntax (latex output)

   647   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

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

   649

   650 translations

   651   "UN x y. B"   == "UN x. UN y. B"

   652   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"

   653   "UN x. B"     == "UN x:CONST UNIV. B"

   654   "UN x:A. B"   == "CONST UNION A (%x. B)"

   655

   656 text {*

   657   Note the difference between ordinary xsymbol syntax of indexed

   658   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})

   659   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The

   660   former does not make the index expression a subscript of the

   661   union/intersection symbol because this leads to problems with nested

   662   subscripts in Proof General.

   663 *}

   664

   665 print_translation {*

   666   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]

   667 *} -- {* to avoid eta-contraction of body *}

   668

   669 lemma UNION_eq_Union_image:

   670   "(\<Union>x\<in>A. B x) = \<Union>(B  A)"

   671   by (fact SUPR_def)

   672

   673 lemma Union_def:

   674   "\<Union>S = (\<Union>x\<in>S. x)"

   675   by (simp add: UNION_eq_Union_image image_def)

   676

   677 lemma UNION_def [no_atp]:

   678   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"

   679   by (auto simp add: UNION_eq_Union_image Union_eq)

   680

   681 lemma Union_image_eq [simp]:

   682   "\<Union>(B  A) = (\<Union>x\<in>A. B x)"

   683   by (rule sym) (fact UNION_eq_Union_image)

   684

   685 lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"

   686   by (unfold UNION_def) blast

   687

   688 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"

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

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

   691   by auto

   692

   693 lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"

   694   by (unfold UNION_def) blast

   695

   696 lemma UN_cong [cong]:

   697     "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"

   698   by (simp add: UNION_def)

   699

   700 lemma strong_UN_cong:

   701     "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"

   702   by (simp add: UNION_def simp_implies_def)

   703

   704 lemma image_eq_UN: "f  A = (\<Union>x\<in>A. {f x})"

   705   by blast

   706

   707 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"

   708   by (fact le_SUPI)

   709

   710 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"

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

   712

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

   714   by blast

   715

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

   717   by blast

   718

   719 lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"

   720   by blast

   721

   722 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"

   723   by blast

   724

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

   726   by blast

   727

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

   729   by auto

   730

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

   732   by blast

   733

   734 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)"

   735   by blast

   736

   737 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)"

   738   by blast

   739

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

   741   by (fact SUP_le_iff)

   742

   743 lemma image_Union: "f  \<Union>S = (\<Union>x\<in>S. f  x)"

   744   by blast

   745

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

   747   by auto

   748

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

   750   by blast

   751

   752 lemma UNION_empty_conv[simp]:

   753   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"

   754   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"

   755 by blast+

   756

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

   758   by blast

   759

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

   761   by blast

   762

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

   764   by blast

   765

   766 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"

   767   by (auto simp add: split_if_mem2)

   768

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

   770   by (auto intro: bool_contrapos)

   771

   772 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"

   773   by blast

   774

   775 lemma UN_mono:

   776   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>

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

   778   by (blast dest: subsetD)

   779

   780 lemma vimage_Union: "f - (\<Union>A) = (\<Union>X\<in>A. f - X)"

   781   by blast

   782

   783 lemma vimage_UN: "f - (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f - B x)"

   784   by blast

   785

   786 lemma vimage_eq_UN: "f - B = (\<Union>y\<in>B. f - {y})"

   787   -- {* NOT suitable for rewriting *}

   788   by blast

   789

   790 lemma image_UN: "f  UNION A B = (\<Union>x\<in>A. f  B x)"

   791   by blast

   792

   793

   794 subsection {* Distributive laws *}

   795

   796 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"

   797   by blast

   798

   799 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"

   800   by blast

   801

   802 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A  C) \<union> \<Union>(B  C)"

   803   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}

   804   -- {* Union of a family of unions *}

   805   by blast

   806

   807 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)"

   808   -- {* Equivalent version *}

   809   by blast

   810

   811 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"

   812   by blast

   813

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

   815   by blast

   816

   817 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)"

   818   -- {* Equivalent version *}

   819   by blast

   820

   821 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"

   822   -- {* Halmos, Naive Set Theory, page 35. *}

   823   by blast

   824

   825 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"

   826   by blast

   827

   828 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)"

   829   by blast

   830

   831 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)"

   832   by blast

   833

   834

   835 subsection {* Complement *}

   836

   837 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"

   838   by blast

   839

   840 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"

   841   by blast

   842

   843

   844 subsection {* Miniscoping and maxiscoping *}

   845

   846 text {* \medskip Miniscoping: pushing in quantifiers and big Unions

   847            and Intersections. *}

   848

   849 lemma UN_simps [simp]:

   850   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"

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

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

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

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

   855   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"

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

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

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

   859   "\<And>A B f. (\<Union>x\<in>fA. B x) = (\<Union>a\<in>A. B (f a))"

   860   by auto

   861

   862 lemma INT_simps [simp]:

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

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

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

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

   867   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"

   868   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"

   869   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"

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

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

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

   873   by auto

   874

   875 lemma ball_simps [simp,no_atp]:

   876   "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"

   877   "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"

   878   "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"

   879   "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"

   880   "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"

   881   "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"

   882   "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"

   883   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"

   884   "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"

   885   "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"

   886   "\<And>A P f. (\<forall>x\<in>fA. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"

   887   "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"

   888   by auto

   889

   890 lemma bex_simps [simp,no_atp]:

   891   "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"

   892   "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"

   893   "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"

   894   "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"

   895   "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"

   896   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"

   897   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"

   898   "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"

   899   "\<And>A P f. (\<exists>x\<in>fA. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"

   900   "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"

   901   by auto

   902

   903 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}

   904

   905 lemma UN_extend_simps:

   906   "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"

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

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

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

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

   911   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"

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

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

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

   915   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>fA. B x)"

   916   by auto

   917

   918 lemma INT_extend_simps:

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

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

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

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

   923   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"

   924   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"

   925   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"

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

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

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

   929   by auto

   930

   931

   932 no_notation

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

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

   935   bot ("\<bottom>") and

   936   top ("\<top>") and

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

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

   939   Inf  ("\<Sqinter>_" [900] 900) and

   940   Sup  ("\<Squnion>_" [900] 900)

   941

   942 no_syntax (xsymbols)

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

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

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

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

   947

   948 lemmas mem_simps =

   949   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

   950   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

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

   952

   953 end
`