src/HOL/Complete_Lattice.thy
 author haftmann Sun Jul 17 19:48:02 2011 +0200 (2011-07-17) changeset 43866 8a50dc70cbff parent 43865 db18f4d0cc7d child 43867 771014555553 permissions -rw-r--r--
moving UNIV = ... equations to their proper theories
     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 INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"

   156   by (simp add: INFI_def)

   157

   158 lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"

   159   by (simp add: INFI_def Inf_insert)

   160

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

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

   163

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

   165   using INF_leI [of i A f] by auto

   166

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

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

   169

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

   171   by (auto simp add: INFI_def le_Inf_iff)

   172

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

   174   by (auto intro: antisym INF_leI le_INFI)

   175

   176 lemma INF_cong:

   177   "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)"

   178   by (simp add: INFI_def image_def)

   179

   180 lemma INF_mono:

   181   "(\<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)"

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

   183

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

   185   by (intro INF_mono) auto

   186

   187 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)"

   188   by (iprover intro: INF_leI le_INFI order_trans antisym)

   189

   190 lemma SUP_cong:

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

   192   by (simp add: SUPR_def image_def)

   193

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

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

   196

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

   198   using le_SUPI [of i A f] by auto

   199

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

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

   202

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

   204   unfolding SUPR_def by (auto simp add: Sup_le_iff)

   205

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

   207   by (auto intro: antisym SUP_leI le_SUPI)

   208

   209 lemma SUP_mono:

   210   "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"

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

   212

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

   214   by (intro SUP_mono) auto

   215

   216 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)"

   217   by (iprover intro: SUP_leI le_SUPI order_trans antisym)

   218

   219 lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"

   220   by (simp add: SUPR_def)

   221

   222 lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"

   223   by (simp add: SUPR_def Sup_insert)

   224

   225 end

   226

   227 lemma Inf_less_iff:

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

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

   230   unfolding not_le [symmetric] le_Inf_iff by auto

   231

   232 lemma INF_less_iff:

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

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

   235   unfolding INFI_def Inf_less_iff by auto

   236

   237 lemma less_Sup_iff:

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

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

   240   unfolding not_le [symmetric] Sup_le_iff by auto

   241

   242 lemma less_SUP_iff:

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

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

   245   unfolding SUPR_def less_Sup_iff by auto

   246

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

   248

   249 instantiation bool :: complete_lattice

   250 begin

   251

   252 definition

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

   254

   255 definition

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

   257

   258 instance proof

   259 qed (auto simp add: Inf_bool_def Sup_bool_def)

   260

   261 end

   262

   263 lemma INFI_bool_eq [simp]:

   264   "INFI = Ball"

   265 proof (rule ext)+

   266   fix A :: "'a set"

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

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

   269     by (auto simp add: Ball_def INFI_def Inf_bool_def)

   270 qed

   271

   272 lemma SUPR_bool_eq [simp]:

   273   "SUPR = Bex"

   274 proof (rule ext)+

   275   fix A :: "'a set"

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

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

   278     by (auto simp add: Bex_def SUPR_def Sup_bool_def)

   279 qed

   280

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

   282 begin

   283

   284 definition

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

   286

   287 lemma Inf_apply:

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

   289   by (simp add: Inf_fun_def)

   290

   291 definition

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

   293

   294 lemma Sup_apply:

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

   296   by (simp add: Sup_fun_def)

   297

   298 instance proof

   299 qed (auto simp add: le_fun_def Inf_apply Sup_apply

   300   intro: Inf_lower Sup_upper Inf_greatest Sup_least)

   301

   302 end

   303

   304 lemma INFI_apply:

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

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

   307

   308 lemma SUPR_apply:

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

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

   311

   312

   313 subsection {* Inter *}

   314

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

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

   317

   318 notation (xsymbols)

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

   320

   321 lemma Inter_eq:

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

   323 proof (rule set_eqI)

   324   fix x

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

   326     by auto

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

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

   329 qed

   330

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

   332   by (unfold Inter_eq) blast

   333

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

   335   by (simp add: Inter_eq)

   336

   337 text {*

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

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

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

   341 *}

   342

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

   344   by auto

   345

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

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

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

   349   by (unfold Inter_eq) blast

   350

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

   352   by (fact Inf_lower)

   353

   354 lemma (in complete_lattice) Inf_less_eq:

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

   356     and "A \<noteq> {}"

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

   358 proof -

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

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

   361   ultimately show ?thesis by (rule Inf_lower2)

   362 qed

   363

   364 lemma Inter_subset:

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

   366   by (fact Inf_less_eq)

   367

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

   369   by (fact Inf_greatest)

   370

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

   372   by (fact Inf_binary [symmetric])

   373

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

   375   by (fact Inf_empty)

   376

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

   378   by (fact Inf_UNIV)

   379

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

   381   by (fact Inf_insert)

   382

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

   384   by (auto intro: Inf_greatest Inf_lower)

   385

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

   387   by (fact Inf_inter_less)

   388

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

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

   391

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

   393   by (fact Inf_union_distrib)

   394

   395 lemma (in complete_lattice) Inf_top_conv [no_atp]:

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

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

   398 proof -

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

   400   proof

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

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

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

   404   next

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

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

   407     proof (rule ccontr)

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

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

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

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

   412     qed

   413   qed

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

   415 qed

   416

   417 lemma Inter_UNIV_conv [simp,no_atp]:

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

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

   420   by (fact Inf_top_conv)+

   421

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

   423   by (auto intro: Inf_greatest Inf_lower)

   424

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

   426   by (fact Inf_anti_mono)

   427

   428

   429 subsection {* Intersections of families *}

   430

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

   432   "INTER \<equiv> INFI"

   433

   434 syntax

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

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

   437

   438 syntax (xsymbols)

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

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

   441

   442 syntax (latex output)

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

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

   445

   446 translations

   447   "INT x y. B"  == "INT x. INT y. B"

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

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

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

   451

   452 print_translation {*

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

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

   455

   456 lemma INTER_eq_Inter_image:

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

   458   by (fact INFI_def)

   459

   460 lemma Inter_def:

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

   462   by (simp add: INTER_eq_Inter_image image_def)

   463

   464 lemma INTER_def:

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

   466   by (auto simp add: INTER_eq_Inter_image Inter_eq)

   467

   468 lemma Inter_image_eq [simp]:

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

   470   by (rule sym) (fact INFI_def)

   471

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

   473   by (unfold INTER_def) blast

   474

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

   476   by (unfold INTER_def) blast

   477

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

   479   by auto

   480

   481 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"

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

   483   by (unfold INTER_def) blast

   484

   485 lemma INT_cong [cong]:

   486   "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)"

   487   by (fact INF_cong)

   488

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

   490   by blast

   491

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

   493   by blast

   494

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

   496   by (fact INF_leI)

   497

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

   499   by (fact le_INFI)

   500

   501 lemma (in complete_lattice) INFI_empty:

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

   503   by (simp add: INFI_def)

   504

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

   506   by (fact INFI_empty)

   507

   508 lemma (in complete_lattice) INFI_absorb:

   509   assumes "k \<in> I"

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

   511 proof -

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

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

   514 qed

   515

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

   517   by (fact INFI_absorb)

   518

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

   520   by (fact le_INF_iff)

   521

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

   523   by (fact INF_insert)

   524

   525 lemma (in complete_lattice) INF_union:

   526   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"

   527   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INFI INF_leI)

   528

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

   530   by (fact INF_union)

   531

   532 lemma INT_insert_distrib:

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

   534   by blast

   535

   536 -- {* continue generalization from here *}

   537

   538 lemma (in complete_lattice) INF_constant:

   539   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"

   540   by (simp add: INF_empty)

   541

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

   543   by (fact INF_constant)

   544

   545 lemma (in complete_lattice) INF_eq:

   546   "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"

   547   by (simp add: INFI_def image_def)

   548

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

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

   551   by (fact INF_eq)

   552

   553 lemma (in complete_lattice) INF_top_conv:

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

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

   556   by (auto simp add: INFI_def Inf_top_conv)

   557

   558 lemma INTER_UNIV_conv [simp]:

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

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

   561   by (fact INF_top_conv)+

   562

   563 lemma (in complete_lattice) INFI_UNIV_range:

   564   "(\<Sqinter>x\<in>UNIV. f x) = \<Sqinter>range f"

   565   by (simp add: INFI_def)

   566

   567 lemma (in complete_lattice) INF_bool_eq:

   568   "(\<Sqinter>b. A b) = A True \<sqinter> A False"

   569   by (simp add: UNIV_bool INF_empty INF_insert inf_commute)

   570

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

   572   by (fact INF_bool_eq)

   573

   574 lemma (in complete_lattice) INF_anti_mono:

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

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

   577   by (blast dest: subsetD)

   578

   579 lemma INT_anti_mono:

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

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

   582   by (blast dest: subsetD)

   583

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

   585   by blast

   586

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

   588   by blast

   589

   590

   591 subsection {* Union *}

   592

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

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

   595

   596 notation (xsymbols)

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

   598

   599 lemma Union_eq:

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

   601 proof (rule set_eqI)

   602   fix x

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

   604     by auto

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

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

   607 qed

   608

   609 lemma Union_iff [simp, no_atp]:

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

   611   by (unfold Union_eq) blast

   612

   613 lemma UnionI [intro]:

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

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

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

   617   by auto

   618

   619 lemma UnionE [elim!]:

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

   621   by auto

   622

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

   624   by (iprover intro: subsetI UnionI)

   625

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

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

   628

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

   630   by blast

   631

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

   633   by blast

   634

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

   636   by blast

   637

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

   639   by blast

   640

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

   642   by blast

   643

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

   645   by blast

   646

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

   648   by blast

   649

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

   651   by blast

   652

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

   654   by blast

   655

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

   657   by blast

   658

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

   660   by blast

   661

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

   663   by blast

   664

   665

   666 subsection {* Unions of families *}

   667

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

   669   "UNION \<equiv> SUPR"

   670

   671 syntax

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

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

   674

   675 syntax (xsymbols)

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

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

   678

   679 syntax (latex output)

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

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

   682

   683 translations

   684   "UN x y. B"   == "UN x. UN y. B"

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

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

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

   688

   689 text {*

   690   Note the difference between ordinary xsymbol syntax of indexed

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

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

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

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

   695   subscripts in Proof General.

   696 *}

   697

   698 print_translation {*

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

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

   701

   702 lemma UNION_eq_Union_image:

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

   704   by (fact SUPR_def)

   705

   706 lemma Union_def:

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

   708   by (simp add: UNION_eq_Union_image image_def)

   709

   710 lemma UNION_def [no_atp]:

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

   712   by (auto simp add: UNION_eq_Union_image Union_eq)

   713

   714 lemma Union_image_eq [simp]:

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

   716   by (rule sym) (fact UNION_eq_Union_image)

   717

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

   719   by (unfold UNION_def) blast

   720

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

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

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

   724   by auto

   725

   726 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"

   727   by (unfold UNION_def) blast

   728

   729 lemma UN_cong [cong]:

   730     "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)"

   731   by (simp add: UNION_def)

   732

   733 lemma strong_UN_cong:

   734     "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)"

   735   by (simp add: UNION_def simp_implies_def)

   736

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

   738   by blast

   739

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

   741   by (fact le_SUPI)

   742

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

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

   745

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

   747   by blast

   748

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

   750   by blast

   751

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

   753   by blast

   754

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

   756   by blast

   757

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

   759   by blast

   760

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

   762   by auto

   763

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

   765   by blast

   766

   767 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)"

   768   by blast

   769

   770 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)"

   771   by blast

   772

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

   774   by (fact SUP_le_iff)

   775

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

   777   by blast

   778

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

   780   by auto

   781

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

   783   by blast

   784

   785 lemma UNION_empty_conv[simp]:

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

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

   788 by blast+

   789

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

   791   by blast

   792

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

   794   by blast

   795

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

   797   by blast

   798

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

   800   by (auto simp add: split_if_mem2)

   801

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

   803   by (auto intro: bool_contrapos)

   804

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

   806   by blast

   807

   808 lemma UN_mono:

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

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

   811   by (blast dest: subsetD)

   812

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

   814   by blast

   815

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

   817   by blast

   818

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

   820   -- {* NOT suitable for rewriting *}

   821   by blast

   822

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

   824   by blast

   825

   826

   827 subsection {* Distributive laws *}

   828

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

   830   by blast

   831

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

   833   by blast

   834

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

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

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

   838   by blast

   839

   840 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)"

   841   -- {* Equivalent version *}

   842   by blast

   843

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

   845   by blast

   846

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

   848   by blast

   849

   850 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)"

   851   -- {* Equivalent version *}

   852   by blast

   853

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

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

   856   by blast

   857

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

   859   by blast

   860

   861 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)"

   862   by blast

   863

   864 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)"

   865   by blast

   866

   867

   868 subsection {* Complement *}

   869

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

   871   by blast

   872

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

   874   by blast

   875

   876

   877 subsection {* Miniscoping and maxiscoping *}

   878

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

   880            and Intersections. *}

   881

   882 lemma UN_simps [simp]:

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

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

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

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

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

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

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

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

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

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

   893   by auto

   894

   895 lemma INT_simps [simp]:

   896   "\<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)"

   897   "\<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))"

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

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

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

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

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

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

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

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

   906   by auto

   907

   908 lemma ball_simps [simp,no_atp]:

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

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

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

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

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

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

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

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

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

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

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

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

   921   by auto

   922

   923 lemma bex_simps [simp,no_atp]:

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

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

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

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

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

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

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

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

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

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

   934   by auto

   935

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

   937

   938 lemma UN_extend_simps:

   939   "\<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)))"

   940   "\<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))"

   941   "\<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))"

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

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

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

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

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

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

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

   949   by auto

   950

   951 lemma INT_extend_simps:

   952   "\<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))"

   953   "\<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))"

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

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

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

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

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

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

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

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

   962   by auto

   963

   964

   965 no_notation

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

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

   968   bot ("\<bottom>") and

   969   top ("\<top>") and

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

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

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

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

   974

   975 no_syntax (xsymbols)

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

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

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

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

   980

   981 lemmas mem_simps =

   982   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

   983   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

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

   985

   986 end
`