src/HOL/Complete_Lattice.thy
author hoelzl
Mon Mar 14 14:37:36 2011 +0100 (2011-03-14)
changeset 41971 a54e8e95fe96
parent 41082 9ff94e7cc3b3
child 42284 326f57825e1a
permissions -rw-r--r--
add lemmas for SUP and INF
     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_empty Inf_insert)
    80 
    81 lemma Sup_binary:
    82   "\<Squnion>{a, b} = a \<squnion> b"
    83   by (simp add: Sup_empty Sup_insert)
    84 
    85 lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
    86   by (auto intro: Inf_greatest dest: Inf_lower)
    87 
    88 lemma Sup_le_iff: "Sup 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 "Inf A \<sqsubseteq> Inf 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 "Inf A \<sqsubseteq> a" by (rule Inf_lower)
    98   with `a \<sqsubseteq> b` show "Inf 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 "Sup A \<sqsubseteq> Sup 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> Sup B" by (rule Sup_upper)
   108   with `a \<sqsubseteq> b` show "a \<sqsubseteq> Sup B" by auto
   109 qed
   110 
   111 lemma top_le:
   112   "top \<sqsubseteq> x \<Longrightarrow> x = top"
   113   by (rule antisym) auto
   114 
   115 lemma le_bot:
   116   "x \<sqsubseteq> bot \<Longrightarrow> x = bot"
   117   by (rule antisym) auto
   118 
   119 lemma not_less_bot[simp]: "\<not> (x \<sqsubset> bot)"
   120   using bot_least[of x] by (auto simp: le_less)
   121 
   122 lemma not_top_less[simp]: "\<not> (top \<sqsubset> x)"
   123   using top_greatest[of x] by (auto simp: le_less)
   124 
   125 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> Sup A"
   126   using Sup_upper[of u A] by auto
   127 
   128 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> Inf A \<sqsubseteq> v"
   129   using Inf_lower[of u A] by auto
   130 
   131 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
   132   "INFI A f = \<Sqinter> (f ` A)"
   133 
   134 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
   135   "SUPR A f = \<Squnion> (f ` A)"
   136 
   137 end
   138 
   139 syntax
   140   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
   141   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
   142   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
   143   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
   144 
   145 syntax (xsymbols)
   146   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
   147   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
   148   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
   149   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
   150 
   151 translations
   152   "INF x y. B"   == "INF x. INF y. B"
   153   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
   154   "INF x. B"     == "INF x:CONST UNIV. B"
   155   "INF x:A. B"   == "CONST INFI A (%x. B)"
   156   "SUP x y. B"   == "SUP x. SUP y. B"
   157   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
   158   "SUP x. B"     == "SUP x:CONST UNIV. B"
   159   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
   160 
   161 print_translation {*
   162   [Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
   163     Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
   164 *} -- {* to avoid eta-contraction of body *}
   165 
   166 context complete_lattice
   167 begin
   168 
   169 lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
   170   by (simp add: SUPR_def cong: image_cong)
   171 
   172 lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
   173   by (simp add: INFI_def cong: image_cong)
   174 
   175 lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"
   176   by (auto simp add: SUPR_def intro: Sup_upper)
   177 
   178 lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (SUP i:A. M i)"
   179   using le_SUPI[of i A M] by auto
   180 
   181 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"
   182   by (auto simp add: SUPR_def intro: Sup_least)
   183 
   184 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"
   185   by (auto simp add: INFI_def intro: Inf_lower)
   186 
   187 lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> u"
   188   using INF_leI[of i A M] by auto
   189 
   190 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
   191   by (auto simp add: INFI_def intro: Inf_greatest)
   192 
   193 lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
   194   unfolding SUPR_def by (auto simp add: Sup_le_iff)
   195 
   196 lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
   197   unfolding INFI_def by (auto simp add: le_Inf_iff)
   198 
   199 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
   200   by (auto intro: antisym INF_leI le_INFI)
   201 
   202 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
   203   by (auto intro: antisym SUP_leI le_SUPI)
   204 
   205 lemma INF_mono:
   206   "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
   207   by (force intro!: Inf_mono simp: INFI_def)
   208 
   209 lemma SUP_mono:
   210   "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)"
   211   by (force intro!: Sup_mono simp: SUPR_def)
   212 
   213 lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
   214   by (intro INF_mono) auto
   215 
   216 lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"
   217   by (intro SUP_mono) auto
   218 
   219 lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
   220   by (iprover intro: INF_leI le_INFI order_trans antisym)
   221 
   222 lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
   223   by (iprover intro: SUP_leI le_SUPI order_trans antisym)
   224 
   225 end
   226 
   227 lemma Inf_less_iff:
   228   fixes a :: "'a\<Colon>{complete_lattice,linorder}"
   229   shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
   230   unfolding not_le[symmetric] le_Inf_iff by auto
   231 
   232 lemma less_Sup_iff:
   233   fixes a :: "'a\<Colon>{complete_lattice,linorder}"
   234   shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
   235   unfolding not_le[symmetric] Sup_le_iff by auto
   236 
   237 lemma INF_less_iff:
   238   fixes a :: "'a::{complete_lattice,linorder}"
   239   shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
   240   unfolding INFI_def Inf_less_iff by auto
   241 
   242 lemma less_SUP_iff:
   243   fixes a :: "'a::{complete_lattice,linorder}"
   244   shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < 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 le_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 "(INF x: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 "(SUP x: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 : Inter C) = (ALL X:C. A:X)"
   332   by (unfold Inter_eq) blast
   333 
   334 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : 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:X"} can hold when
   340   @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
   341 *}
   342 
   343 lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
   344   by auto
   345 
   346 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
   347   -- {* ``Classical'' elimination rule -- does not require proving
   348     @{prop "X:C"}. *}
   349   by (unfold Inter_eq) blast
   350 
   351 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
   352   by blast
   353 
   354 lemma Inter_subset:
   355   "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
   356   by blast
   357 
   358 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
   359   by (iprover intro: InterI subsetI dest: subsetD)
   360 
   361 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
   362   by blast
   363 
   364 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
   365   by (fact Inf_empty)
   366 
   367 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
   368   by blast
   369 
   370 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
   371   by blast
   372 
   373 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
   374   by blast
   375 
   376 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
   377   by blast
   378 
   379 lemma Inter_UNIV_conv [simp,no_atp]:
   380   "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
   381   "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
   382   by blast+
   383 
   384 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
   385   by blast
   386 
   387 
   388 subsection {* Intersections of families *}
   389 
   390 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
   391   "INTER \<equiv> INFI"
   392 
   393 syntax
   394   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
   395   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
   396 
   397 syntax (xsymbols)
   398   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
   399   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
   400 
   401 syntax (latex output)
   402   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
   403   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
   404 
   405 translations
   406   "INT x y. B"  == "INT x. INT y. B"
   407   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
   408   "INT x. B"    == "INT x:CONST UNIV. B"
   409   "INT x:A. B"  == "CONST INTER A (%x. B)"
   410 
   411 print_translation {*
   412   [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
   413 *} -- {* to avoid eta-contraction of body *}
   414 
   415 lemma INTER_eq_Inter_image:
   416   "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
   417   by (fact INFI_def)
   418   
   419 lemma Inter_def:
   420   "\<Inter>S = (\<Inter>x\<in>S. x)"
   421   by (simp add: INTER_eq_Inter_image image_def)
   422 
   423 lemma INTER_def:
   424   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
   425   by (auto simp add: INTER_eq_Inter_image Inter_eq)
   426 
   427 lemma Inter_image_eq [simp]:
   428   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
   429   by (rule sym) (fact INTER_eq_Inter_image)
   430 
   431 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
   432   by (unfold INTER_def) blast
   433 
   434 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
   435   by (unfold INTER_def) blast
   436 
   437 lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
   438   by auto
   439 
   440 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
   441   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
   442   by (unfold INTER_def) blast
   443 
   444 lemma INT_cong [cong]:
   445     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
   446   by (simp add: INTER_def)
   447 
   448 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
   449   by blast
   450 
   451 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
   452   by blast
   453 
   454 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
   455   by (fact INF_leI)
   456 
   457 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
   458   by (fact le_INFI)
   459 
   460 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
   461   by blast
   462 
   463 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
   464   by blast
   465 
   466 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
   467   by (fact le_INF_iff)
   468 
   469 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
   470   by blast
   471 
   472 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
   473   by blast
   474 
   475 lemma INT_insert_distrib:
   476     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
   477   by blast
   478 
   479 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
   480   by auto
   481 
   482 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
   483   -- {* Look: it has an \emph{existential} quantifier *}
   484   by blast
   485 
   486 lemma INTER_UNIV_conv[simp]:
   487  "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
   488  "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
   489 by blast+
   490 
   491 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
   492   by (auto intro: bool_induct)
   493 
   494 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
   495   by blast
   496 
   497 lemma INT_anti_mono:
   498   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
   499     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
   500   -- {* The last inclusion is POSITIVE! *}
   501   by (blast dest: subsetD)
   502 
   503 lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
   504   by blast
   505 
   506 
   507 subsection {* Union *}
   508 
   509 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
   510   "Union S \<equiv> \<Squnion>S"
   511 
   512 notation (xsymbols)
   513   Union  ("\<Union>_" [90] 90)
   514 
   515 lemma Union_eq:
   516   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
   517 proof (rule set_eqI)
   518   fix x
   519   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
   520     by auto
   521   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
   522     by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
   523 qed
   524 
   525 lemma Union_iff [simp, no_atp]:
   526   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
   527   by (unfold Union_eq) blast
   528 
   529 lemma UnionI [intro]:
   530   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
   531   -- {* The order of the premises presupposes that @{term C} is rigid;
   532     @{term A} may be flexible. *}
   533   by auto
   534 
   535 lemma UnionE [elim!]:
   536   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
   537   by auto
   538 
   539 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
   540   by (iprover intro: subsetI UnionI)
   541 
   542 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
   543   by (iprover intro: subsetI elim: UnionE dest: subsetD)
   544 
   545 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
   546   by blast
   547 
   548 lemma Union_empty [simp]: "Union({}) = {}"
   549   by blast
   550 
   551 lemma Union_UNIV [simp]: "Union UNIV = UNIV"
   552   by blast
   553 
   554 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
   555   by blast
   556 
   557 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
   558   by blast
   559 
   560 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
   561   by blast
   562 
   563 lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
   564   by blast
   565 
   566 lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
   567   by blast
   568 
   569 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
   570   by blast
   571 
   572 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
   573   by blast
   574 
   575 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
   576   by blast
   577 
   578 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
   579   by blast
   580 
   581 
   582 subsection {* Unions of families *}
   583 
   584 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
   585   "UNION \<equiv> SUPR"
   586 
   587 syntax
   588   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
   589   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
   590 
   591 syntax (xsymbols)
   592   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
   593   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
   594 
   595 syntax (latex output)
   596   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
   597   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
   598 
   599 translations
   600   "UN x y. B"   == "UN x. UN y. B"
   601   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
   602   "UN x. B"     == "UN x:CONST UNIV. B"
   603   "UN x:A. B"   == "CONST UNION A (%x. B)"
   604 
   605 text {*
   606   Note the difference between ordinary xsymbol syntax of indexed
   607   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
   608   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
   609   former does not make the index expression a subscript of the
   610   union/intersection symbol because this leads to problems with nested
   611   subscripts in Proof General.
   612 *}
   613 
   614 print_translation {*
   615   [Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
   616 *} -- {* to avoid eta-contraction of body *}
   617 
   618 lemma UNION_eq_Union_image:
   619   "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
   620   by (fact SUPR_def)
   621 
   622 lemma Union_def:
   623   "\<Union>S = (\<Union>x\<in>S. x)"
   624   by (simp add: UNION_eq_Union_image image_def)
   625 
   626 lemma UNION_def [no_atp]:
   627   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
   628   by (auto simp add: UNION_eq_Union_image Union_eq)
   629   
   630 lemma Union_image_eq [simp]:
   631   "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
   632   by (rule sym) (fact UNION_eq_Union_image)
   633   
   634 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
   635   by (unfold UNION_def) blast
   636 
   637 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
   638   -- {* The order of the premises presupposes that @{term A} is rigid;
   639     @{term b} may be flexible. *}
   640   by auto
   641 
   642 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
   643   by (unfold UNION_def) blast
   644 
   645 lemma UN_cong [cong]:
   646     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
   647   by (simp add: UNION_def)
   648 
   649 lemma strong_UN_cong:
   650     "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
   651   by (simp add: UNION_def simp_implies_def)
   652 
   653 lemma image_eq_UN: "f`A = (UN x:A. {f x})"
   654   by blast
   655 
   656 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
   657   by (fact le_SUPI)
   658 
   659 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
   660   by (iprover intro: subsetI elim: UN_E dest: subsetD)
   661 
   662 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
   663   by blast
   664 
   665 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
   666   by blast
   667 
   668 lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
   669   by blast
   670 
   671 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
   672   by blast
   673 
   674 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
   675   by blast
   676 
   677 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
   678   by auto
   679 
   680 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
   681   by blast
   682 
   683 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)"
   684   by blast
   685 
   686 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)"
   687   by blast
   688 
   689 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
   690   by (fact SUP_le_iff)
   691 
   692 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
   693   by blast
   694 
   695 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
   696   by auto
   697 
   698 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
   699   by blast
   700 
   701 lemma UNION_empty_conv[simp]:
   702   "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
   703   "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
   704 by blast+
   705 
   706 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
   707   by blast
   708 
   709 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
   710   by blast
   711 
   712 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
   713   by blast
   714 
   715 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
   716   by (auto simp add: split_if_mem2)
   717 
   718 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
   719   by (auto intro: bool_contrapos)
   720 
   721 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
   722   by blast
   723 
   724 lemma UN_mono:
   725   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
   726     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
   727   by (blast dest: subsetD)
   728 
   729 lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
   730   by blast
   731 
   732 lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
   733   by blast
   734 
   735 lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
   736   -- {* NOT suitable for rewriting *}
   737   by blast
   738 
   739 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
   740 by blast
   741 
   742 
   743 subsection {* Distributive laws *}
   744 
   745 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
   746   by blast
   747 
   748 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
   749   by blast
   750 
   751 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
   752   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
   753   -- {* Union of a family of unions *}
   754   by blast
   755 
   756 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)"
   757   -- {* Equivalent version *}
   758   by blast
   759 
   760 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
   761   by blast
   762 
   763 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
   764   by blast
   765 
   766 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)"
   767   -- {* Equivalent version *}
   768   by blast
   769 
   770 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
   771   -- {* Halmos, Naive Set Theory, page 35. *}
   772   by blast
   773 
   774 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
   775   by blast
   776 
   777 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)"
   778   by blast
   779 
   780 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)"
   781   by blast
   782 
   783 
   784 subsection {* Complement *}
   785 
   786 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
   787   by blast
   788 
   789 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
   790   by blast
   791 
   792 
   793 subsection {* Miniscoping and maxiscoping *}
   794 
   795 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
   796            and Intersections. *}
   797 
   798 lemma UN_simps [simp]:
   799   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
   800   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
   801   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
   802   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
   803   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
   804   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
   805   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
   806   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
   807   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
   808   "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
   809   by auto
   810 
   811 lemma INT_simps [simp]:
   812   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
   813   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
   814   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
   815   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
   816   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
   817   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
   818   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
   819   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
   820   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
   821   "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
   822   by auto
   823 
   824 lemma ball_simps [simp,no_atp]:
   825   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
   826   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
   827   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
   828   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
   829   "!!P. (ALL x:{}. P x) = True"
   830   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
   831   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
   832   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
   833   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
   834   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
   835   "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
   836   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
   837   by auto
   838 
   839 lemma bex_simps [simp,no_atp]:
   840   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
   841   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
   842   "!!P. (EX x:{}. P x) = False"
   843   "!!P. (EX x:UNIV. P x) = (EX x. P x)"
   844   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
   845   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
   846   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
   847   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
   848   "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
   849   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
   850   by auto
   851 
   852 lemma ball_conj_distrib:
   853   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
   854   by blast
   855 
   856 lemma bex_disj_distrib:
   857   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
   858   by blast
   859 
   860 
   861 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
   862 
   863 lemma UN_extend_simps:
   864   "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
   865   "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
   866   "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
   867   "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
   868   "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
   869   "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
   870   "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
   871   "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
   872   "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
   873   "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
   874   by auto
   875 
   876 lemma INT_extend_simps:
   877   "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
   878   "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
   879   "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
   880   "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
   881   "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
   882   "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
   883   "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
   884   "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
   885   "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
   886   "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
   887   by auto
   888 
   889 
   890 no_notation
   891   less_eq  (infix "\<sqsubseteq>" 50) and
   892   less (infix "\<sqsubset>" 50) and
   893   bot ("\<bottom>") and
   894   top ("\<top>") and
   895   inf  (infixl "\<sqinter>" 70) and
   896   sup  (infixl "\<squnion>" 65) and
   897   Inf  ("\<Sqinter>_" [900] 900) and
   898   Sup  ("\<Squnion>_" [900] 900)
   899 
   900 no_syntax (xsymbols)
   901   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
   902   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
   903   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
   904   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
   905 
   906 lemmas mem_simps =
   907   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
   908   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
   909   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
   910 
   911 end