src/HOL/Complete_Lattice.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 44104 50c067b51135 child 44322 43b465f4c480 permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1  (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Complete lattices *}
```
```     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 >) sup inf \<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 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    42   INF_def: "INFI A f = \<Sqinter>(f ` A)"
```
```    43
```
```    44 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    45   SUP_def: "SUPR A f = \<Squnion>(f ` A)"
```
```    46
```
```    47 text {*
```
```    48   Note: must use names @{const INFI} and @{const SUPR} here instead of
```
```    49   @{text INF} and @{text SUP} to allow the following syntax coexist
```
```    50   with the plain constant names.
```
```    51 *}
```
```    52
```
```    53 end
```
```    54
```
```    55 syntax
```
```    56   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```    57   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
```
```    58   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```    59   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
```
```    60
```
```    61 syntax (xsymbols)
```
```    62   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
```
```    63   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
```
```    64   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
```
```    65   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
```
```    66
```
```    67 translations
```
```    68   "INF x y. B"   == "INF x. INF y. B"
```
```    69   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
```
```    70   "INF x. B"     == "INF x:CONST UNIV. B"
```
```    71   "INF x:A. B"   == "CONST INFI A (%x. B)"
```
```    72   "SUP x y. B"   == "SUP x. SUP y. B"
```
```    73   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
```
```    74   "SUP x. B"     == "SUP x:CONST UNIV. B"
```
```    75   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
```
```    76
```
```    77 print_translation {*
```
```    78   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
```
```    79     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
```
```    80 *} -- {* to avoid eta-contraction of body *}
```
```    81
```
```    82 context complete_lattice
```
```    83 begin
```
```    84
```
```    85 lemma INF_foundation_dual [no_atp]:
```
```    86   "complete_lattice.SUPR Inf = INFI"
```
```    87 proof (rule ext)+
```
```    88   interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
```
```    89     by (fact dual_complete_lattice)
```
```    90   fix f :: "'b \<Rightarrow> 'a" and A
```
```    91   show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
```
```    92     by (simp only: dual.SUP_def INF_def)
```
```    93 qed
```
```    94
```
```    95 lemma SUP_foundation_dual [no_atp]:
```
```    96   "complete_lattice.INFI Sup = SUPR"
```
```    97 proof (rule ext)+
```
```    98   interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
```
```    99     by (fact dual_complete_lattice)
```
```   100   fix f :: "'b \<Rightarrow> 'a" and A
```
```   101   show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
```
```   102     by (simp only: dual.INF_def SUP_def)
```
```   103 qed
```
```   104
```
```   105 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
```
```   106   by (auto simp add: INF_def intro: Inf_lower)
```
```   107
```
```   108 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
```
```   109   by (auto simp add: INF_def intro: Inf_greatest)
```
```   110
```
```   111 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   112   by (auto simp add: SUP_def intro: Sup_upper)
```
```   113
```
```   114 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
```
```   115   by (auto simp add: SUP_def intro: Sup_least)
```
```   116
```
```   117 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
```
```   118   using Inf_lower [of u A] by auto
```
```   119
```
```   120 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
```
```   121   using INF_lower [of i A f] by auto
```
```   122
```
```   123 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
```
```   124   using Sup_upper [of u A] by auto
```
```   125
```
```   126 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   127   using SUP_upper [of i A f] by auto
```
```   128
```
```   129 lemma le_Inf_iff (*[simp]*): "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
```
```   130   by (auto intro: Inf_greatest dest: Inf_lower)
```
```   131
```
```   132 lemma le_INF_iff (*[simp]*): "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
```
```   133   by (auto simp add: INF_def le_Inf_iff)
```
```   134
```
```   135 lemma Sup_le_iff (*[simp]*): "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
```
```   136   by (auto intro: Sup_least dest: Sup_upper)
```
```   137
```
```   138 lemma SUP_le_iff (*[simp]*): "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
```
```   139   by (auto simp add: SUP_def Sup_le_iff)
```
```   140
```
```   141 lemma Inf_empty [simp]:
```
```   142   "\<Sqinter>{} = \<top>"
```
```   143   by (auto intro: antisym Inf_greatest)
```
```   144
```
```   145 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
```
```   146   by (simp add: INF_def)
```
```   147
```
```   148 lemma Sup_empty [simp]:
```
```   149   "\<Squnion>{} = \<bottom>"
```
```   150   by (auto intro: antisym Sup_least)
```
```   151
```
```   152 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
```
```   153   by (simp add: SUP_def)
```
```   154
```
```   155 lemma Inf_UNIV [simp]:
```
```   156   "\<Sqinter>UNIV = \<bottom>"
```
```   157   by (auto intro!: antisym Inf_lower)
```
```   158
```
```   159 lemma Sup_UNIV [simp]:
```
```   160   "\<Squnion>UNIV = \<top>"
```
```   161   by (auto intro!: antisym Sup_upper)
```
```   162
```
```   163 lemma Inf_insert (*[simp]*): "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   164   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
```
```   165
```
```   166 lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
```
```   167   by (simp add: INF_def Inf_insert)
```
```   168
```
```   169 lemma Sup_insert (*[simp]*): "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   170   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
```
```   171
```
```   172 lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
```
```   173   by (simp add: SUP_def Sup_insert)
```
```   174
```
```   175 lemma INF_image (*[simp]*): "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
```
```   176   by (simp add: INF_def image_image)
```
```   177
```
```   178 lemma SUP_image (*[simp]*): "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
```
```   179   by (simp add: SUP_def image_image)
```
```   180
```
```   181 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
```
```   182   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   183
```
```   184 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
```
```   185   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   186
```
```   187 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   188   by (auto intro: Inf_greatest Inf_lower)
```
```   189
```
```   190 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   191   by (auto intro: Sup_least Sup_upper)
```
```   192
```
```   193 lemma INF_cong:
```
```   194   "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)"
```
```   195   by (simp add: INF_def image_def)
```
```   196
```
```   197 lemma SUP_cong:
```
```   198   "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)"
```
```   199   by (simp add: SUP_def image_def)
```
```   200
```
```   201 lemma Inf_mono:
```
```   202   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
```
```   203   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   204 proof (rule Inf_greatest)
```
```   205   fix b assume "b \<in> B"
```
```   206   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
```
```   207   from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
```
```   208   with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
```
```   209 qed
```
```   210
```
```   211 lemma INF_mono:
```
```   212   "(\<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)"
```
```   213   by (force intro!: Inf_mono simp: INF_def)
```
```   214
```
```   215 lemma Sup_mono:
```
```   216   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
```
```   217   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   218 proof (rule Sup_least)
```
```   219   fix a assume "a \<in> A"
```
```   220   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
```
```   221   from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
```
```   222   with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
```
```   223 qed
```
```   224
```
```   225 lemma SUP_mono:
```
```   226   "(\<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)"
```
```   227   by (force intro!: Sup_mono simp: SUP_def)
```
```   228
```
```   229 lemma INF_superset_mono:
```
```   230   "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
```
```   231   -- {* The last inclusion is POSITIVE! *}
```
```   232   by (blast intro: INF_mono dest: subsetD)
```
```   233
```
```   234 lemma SUP_subset_mono:
```
```   235   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
```
```   236   by (blast intro: SUP_mono dest: subsetD)
```
```   237
```
```   238 lemma Inf_less_eq:
```
```   239   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
```
```   240     and "A \<noteq> {}"
```
```   241   shows "\<Sqinter>A \<sqsubseteq> u"
```
```   242 proof -
```
```   243   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   244   moreover with assms have "v \<sqsubseteq> u" by blast
```
```   245   ultimately show ?thesis by (rule Inf_lower2)
```
```   246 qed
```
```   247
```
```   248 lemma less_eq_Sup:
```
```   249   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
```
```   250     and "A \<noteq> {}"
```
```   251   shows "u \<sqsubseteq> \<Squnion>A"
```
```   252 proof -
```
```   253   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   254   moreover with assms have "u \<sqsubseteq> v" by blast
```
```   255   ultimately show ?thesis by (rule Sup_upper2)
```
```   256 qed
```
```   257
```
```   258 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
```
```   259   by (auto intro: Inf_greatest Inf_lower)
```
```   260
```
```   261 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
```
```   262   by (auto intro: Sup_least Sup_upper)
```
```   263
```
```   264 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
```
```   265   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
```
```   266
```
```   267 lemma INF_union:
```
```   268   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
```
```   269   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
```
```   270
```
```   271 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
```
```   272   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
```
```   273
```
```   274 lemma SUP_union:
```
```   275   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
```
```   276   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
```
```   277
```
```   278 lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
```
```   279   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
```
```   280
```
```   281 lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)"
```
```   282   by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
```
```   283     rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
```
```   284
```
```   285 lemma Inf_top_conv (*[simp]*) [no_atp]:
```
```   286   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   287   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   288 proof -
```
```   289   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   290   proof
```
```   291     assume "\<forall>x\<in>A. x = \<top>"
```
```   292     then have "A = {} \<or> A = {\<top>}" by auto
```
```   293     then show "\<Sqinter>A = \<top>" by (auto simp add: Inf_insert)
```
```   294   next
```
```   295     assume "\<Sqinter>A = \<top>"
```
```   296     show "\<forall>x\<in>A. x = \<top>"
```
```   297     proof (rule ccontr)
```
```   298       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
```
```   299       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
```
```   300       then obtain B where "A = insert x B" by blast
```
```   301       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
```
```   302     qed
```
```   303   qed
```
```   304   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
```
```   305 qed
```
```   306
```
```   307 lemma INF_top_conv (*[simp]*):
```
```   308  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   309  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   310   by (auto simp add: INF_def Inf_top_conv)
```
```   311
```
```   312 lemma Sup_bot_conv (*[simp]*) [no_atp]:
```
```   313   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
```
```   314   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
```
```   315 proof -
```
```   316   interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
```
```   317     by (fact dual_complete_lattice)
```
```   318   from dual.Inf_top_conv show ?P and ?Q by simp_all
```
```   319 qed
```
```   320
```
```   321 lemma SUP_bot_conv (*[simp]*):
```
```   322  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   323  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   324   by (auto simp add: SUP_def Sup_bot_conv)
```
```   325
```
```   326 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
```
```   327   by (auto intro: antisym INF_lower INF_greatest)
```
```   328
```
```   329 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
```
```   330   by (auto intro: antisym SUP_upper SUP_least)
```
```   331
```
```   332 lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
```
```   333   by (cases "A = {}") (simp_all add: INF_empty)
```
```   334
```
```   335 lemma SUP_bot (*[simp]*): "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
```
```   336   by (cases "A = {}") (simp_all add: SUP_empty)
```
```   337
```
```   338 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)"
```
```   339   by (iprover intro: INF_lower INF_greatest order_trans antisym)
```
```   340
```
```   341 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)"
```
```   342   by (iprover intro: SUP_upper SUP_least order_trans antisym)
```
```   343
```
```   344 lemma INF_absorb:
```
```   345   assumes "k \<in> I"
```
```   346   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
```
```   347 proof -
```
```   348   from assms obtain J where "I = insert k J" by blast
```
```   349   then show ?thesis by (simp add: INF_insert)
```
```   350 qed
```
```   351
```
```   352 lemma SUP_absorb:
```
```   353   assumes "k \<in> I"
```
```   354   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
```
```   355 proof -
```
```   356   from assms obtain J where "I = insert k J" by blast
```
```   357   then show ?thesis by (simp add: SUP_insert)
```
```   358 qed
```
```   359
```
```   360 lemma INF_constant:
```
```   361   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
```
```   362   by (simp add: INF_empty)
```
```   363
```
```   364 lemma SUP_constant:
```
```   365   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
```
```   366   by (simp add: SUP_empty)
```
```   367
```
```   368 lemma less_INF_D:
```
```   369   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
```
```   370 proof -
```
```   371   note `y < (\<Sqinter>i\<in>A. f i)`
```
```   372   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
```
```   373     by (rule INF_lower)
```
```   374   finally show "y < f i" .
```
```   375 qed
```
```   376
```
```   377 lemma SUP_lessD:
```
```   378   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
```
```   379 proof -
```
```   380   have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
```
```   381     by (rule SUP_upper)
```
```   382   also note `(\<Squnion>i\<in>A. f i) < y`
```
```   383   finally show "f i < y" .
```
```   384 qed
```
```   385
```
```   386 lemma INF_UNIV_bool_expand:
```
```   387   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
```
```   388   by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
```
```   389
```
```   390 lemma SUP_UNIV_bool_expand:
```
```   391   "(\<Squnion>b. A b) = A True \<squnion> A False"
```
```   392   by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
```
```   393
```
```   394 end
```
```   395
```
```   396 class complete_distrib_lattice = complete_lattice +
```
```   397   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
```
```   398   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   399 begin
```
```   400
```
```   401 lemma sup_INF:
```
```   402   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
```
```   403   by (simp add: INF_def sup_Inf image_image)
```
```   404
```
```   405 lemma inf_SUP:
```
```   406   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
```
```   407   by (simp add: SUP_def inf_Sup image_image)
```
```   408
```
```   409 lemma dual_complete_distrib_lattice:
```
```   410   "class.complete_distrib_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
```
```   411   apply (rule class.complete_distrib_lattice.intro)
```
```   412   apply (fact dual_complete_lattice)
```
```   413   apply (rule class.complete_distrib_lattice_axioms.intro)
```
```   414   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
```
```   415   done
```
```   416
```
```   417 subclass distrib_lattice proof -- {* Question: is it sufficient to include @{class distrib_lattice}
```
```   418   and prove @{text inf_Sup} and @{text sup_Inf} from that? *}
```
```   419   fix a b c
```
```   420   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
```
```   421   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
```
```   422 qed
```
```   423
```
```   424 lemma Inf_sup:
```
```   425   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
```
```   426   by (simp add: sup_Inf sup_commute)
```
```   427
```
```   428 lemma Sup_inf:
```
```   429   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
```
```   430   by (simp add: inf_Sup inf_commute)
```
```   431
```
```   432 lemma INF_sup:
```
```   433   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
```
```   434   by (simp add: sup_INF sup_commute)
```
```   435
```
```   436 lemma SUP_inf:
```
```   437   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
```
```   438   by (simp add: inf_SUP inf_commute)
```
```   439
```
```   440 lemma Inf_sup_eq_top_iff:
```
```   441   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
```
```   442   by (simp only: Inf_sup INF_top_conv)
```
```   443
```
```   444 lemma Sup_inf_eq_bot_iff:
```
```   445   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
```
```   446   by (simp only: Sup_inf SUP_bot_conv)
```
```   447
```
```   448 lemma INF_sup_distrib2:
```
```   449   "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
```
```   450   by (subst INF_commute) (simp add: sup_INF INF_sup)
```
```   451
```
```   452 lemma SUP_inf_distrib2:
```
```   453   "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
```
```   454   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
```
```   455
```
```   456 end
```
```   457
```
```   458 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
```
```   459 begin
```
```   460
```
```   461 lemma dual_complete_boolean_algebra:
```
```   462   "class.complete_boolean_algebra Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
```
```   463   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
```
```   464
```
```   465 lemma uminus_Inf:
```
```   466   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
```
```   467 proof (rule antisym)
```
```   468   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
```
```   469     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
```
```   470   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
```
```   471     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
```
```   472 qed
```
```   473
```
```   474 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
```
```   475   by (simp add: INF_def SUP_def uminus_Inf image_image)
```
```   476
```
```   477 lemma uminus_Sup:
```
```   478   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
```
```   479 proof -
```
```   480   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
```
```   481   then show ?thesis by simp
```
```   482 qed
```
```   483
```
```   484 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
```
```   485   by (simp add: INF_def SUP_def uminus_Sup image_image)
```
```   486
```
```   487 end
```
```   488
```
```   489 class complete_linorder = linorder + complete_lattice
```
```   490 begin
```
```   491
```
```   492 lemma dual_complete_linorder:
```
```   493   "class.complete_linorder Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
```
```   494   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
```
```   495
```
```   496 lemma Inf_less_iff (*[simp]*):
```
```   497   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
```
```   498   unfolding not_le [symmetric] le_Inf_iff by auto
```
```   499
```
```   500 lemma INF_less_iff (*[simp]*):
```
```   501   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
```
```   502   unfolding INF_def Inf_less_iff by auto
```
```   503
```
```   504 lemma less_Sup_iff (*[simp]*):
```
```   505   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
```
```   506   unfolding not_le [symmetric] Sup_le_iff by auto
```
```   507
```
```   508 lemma less_SUP_iff (*[simp]*):
```
```   509   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
```
```   510   unfolding SUP_def less_Sup_iff by auto
```
```   511
```
```   512 lemma Sup_eq_top_iff (*[simp]*):
```
```   513   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
```
```   514 proof
```
```   515   assume *: "\<Squnion>A = \<top>"
```
```   516   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
```
```   517   proof (intro allI impI)
```
```   518     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
```
```   519       unfolding less_Sup_iff by auto
```
```   520   qed
```
```   521 next
```
```   522   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
```
```   523   show "\<Squnion>A = \<top>"
```
```   524   proof (rule ccontr)
```
```   525     assume "\<Squnion>A \<noteq> \<top>"
```
```   526     with top_greatest [of "\<Squnion>A"]
```
```   527     have "\<Squnion>A < \<top>" unfolding le_less by auto
```
```   528     then have "\<Squnion>A < \<Squnion>A"
```
```   529       using * unfolding less_Sup_iff by auto
```
```   530     then show False by auto
```
```   531   qed
```
```   532 qed
```
```   533
```
```   534 lemma SUP_eq_top_iff (*[simp]*):
```
```   535   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
```
```   536   unfolding SUP_def Sup_eq_top_iff by auto
```
```   537
```
```   538 lemma Inf_eq_bot_iff (*[simp]*):
```
```   539   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
```
```   540 proof -
```
```   541   interpret dual: complete_linorder Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
```
```   542     by (fact dual_complete_linorder)
```
```   543   from dual.Sup_eq_top_iff show ?thesis .
```
```   544 qed
```
```   545
```
```   546 lemma INF_eq_bot_iff (*[simp]*):
```
```   547   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
```
```   548   unfolding INF_def Inf_eq_bot_iff by auto
```
```   549
```
```   550 end
```
```   551
```
```   552
```
```   553 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
```
```   554
```
```   555 instantiation bool :: complete_lattice
```
```   556 begin
```
```   557
```
```   558 definition
```
```   559   "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
```
```   560
```
```   561 definition
```
```   562   "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
```
```   563
```
```   564 instance proof
```
```   565 qed (auto simp add: Inf_bool_def Sup_bool_def)
```
```   566
```
```   567 end
```
```   568
```
```   569 lemma INF_bool_eq [simp]:
```
```   570   "INFI = Ball"
```
```   571 proof (rule ext)+
```
```   572   fix A :: "'a set"
```
```   573   fix P :: "'a \<Rightarrow> bool"
```
```   574   show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
```
```   575     by (auto simp add: Ball_def INF_def Inf_bool_def)
```
```   576 qed
```
```   577
```
```   578 lemma SUP_bool_eq [simp]:
```
```   579   "SUPR = Bex"
```
```   580 proof (rule ext)+
```
```   581   fix A :: "'a set"
```
```   582   fix P :: "'a \<Rightarrow> bool"
```
```   583   show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
```
```   584     by (auto simp add: Bex_def SUP_def Sup_bool_def)
```
```   585 qed
```
```   586
```
```   587 instance bool :: complete_boolean_algebra proof
```
```   588 qed (auto simp add: Inf_bool_def Sup_bool_def)
```
```   589
```
```   590 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   591 begin
```
```   592
```
```   593 definition
```
```   594   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
```
```   595
```
```   596 lemma Inf_apply:
```
```   597   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
```
```   598   by (simp add: Inf_fun_def)
```
```   599
```
```   600 definition
```
```   601   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
```
```   602
```
```   603 lemma Sup_apply:
```
```   604   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
```
```   605   by (simp add: Sup_fun_def)
```
```   606
```
```   607 instance proof
```
```   608 qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
```
```   609
```
```   610 end
```
```   611
```
```   612 lemma INF_apply:
```
```   613   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
```
```   614   by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
```
```   615
```
```   616 lemma SUP_apply:
```
```   617   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
```
```   618   by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
```
```   619
```
```   620 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
```
```   621 qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
```
```   622
```
```   623 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
```
```   624
```
```   625
```
```   626 subsection {* Inter *}
```
```   627
```
```   628 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
```
```   629   "Inter S \<equiv> \<Sqinter>S"
```
```   630
```
```   631 notation (xsymbols)
```
```   632   Inter  ("\<Inter>_" [90] 90)
```
```   633
```
```   634 lemma Inter_eq:
```
```   635   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
```
```   636 proof (rule set_eqI)
```
```   637   fix x
```
```   638   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
```
```   639     by auto
```
```   640   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
```
```   641     by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
```
```   642 qed
```
```   643
```
```   644 lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
```
```   645   by (unfold Inter_eq) blast
```
```   646
```
```   647 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
```
```   648   by (simp add: Inter_eq)
```
```   649
```
```   650 text {*
```
```   651   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
```
```   652   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
```
```   653   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
```
```   654 *}
```
```   655
```
```   656 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
```
```   657   by auto
```
```   658
```
```   659 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
```
```   660   -- {* ``Classical'' elimination rule -- does not require proving
```
```   661     @{prop "X \<in> C"}. *}
```
```   662   by (unfold Inter_eq) blast
```
```   663
```
```   664 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   665   by (fact Inf_lower)
```
```   666
```
```   667 lemma Inter_subset:
```
```   668   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   669   by (fact Inf_less_eq)
```
```   670
```
```   671 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
```
```   672   by (fact Inf_greatest)
```
```   673
```
```   674 lemma Inter_empty: "\<Inter>{} = UNIV"
```
```   675   by (fact Inf_empty) (* already simp *)
```
```   676
```
```   677 lemma Inter_UNIV: "\<Inter>UNIV = {}"
```
```   678   by (fact Inf_UNIV) (* already simp *)
```
```   679
```
```   680 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
```
```   681   by (fact Inf_insert)
```
```   682
```
```   683 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
```
```   684   by (fact less_eq_Inf_inter)
```
```   685
```
```   686 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
```
```   687   by (fact Inf_union_distrib)
```
```   688
```
```   689 lemma Inter_UNIV_conv [simp, no_atp]:
```
```   690   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   691   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   692   by (fact Inf_top_conv)+
```
```   693
```
```   694 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
```
```   695   by (fact Inf_superset_mono)
```
```   696
```
```   697
```
```   698 subsection {* Intersections of families *}
```
```   699
```
```   700 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```   701   "INTER \<equiv> INFI"
```
```   702
```
```   703 text {*
```
```   704   Note: must use name @{const INTER} here instead of @{text INT}
```
```   705   to allow the following syntax coexist with the plain constant name.
```
```   706 *}
```
```   707
```
```   708 syntax
```
```   709   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
```
```   710   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
```
```   711
```
```   712 syntax (xsymbols)
```
```   713   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
```
```   714   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
```
```   715
```
```   716 syntax (latex output)
```
```   717   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   718   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   719
```
```   720 translations
```
```   721   "INT x y. B"  == "INT x. INT y. B"
```
```   722   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
```
```   723   "INT x. B"    == "INT x:CONST UNIV. B"
```
```   724   "INT x:A. B"  == "CONST INTER A (%x. B)"
```
```   725
```
```   726 print_translation {*
```
```   727   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
```
```   728 *} -- {* to avoid eta-contraction of body *}
```
```   729
```
```   730 lemma INTER_eq:
```
```   731   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
```
```   732   by (auto simp add: INF_def)
```
```   733
```
```   734 lemma Inter_image_eq [simp]:
```
```   735   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
```
```   736   by (rule sym) (fact INF_def)
```
```   737
```
```   738 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
```
```   739   by (auto simp add: INF_def image_def)
```
```   740
```
```   741 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
```
```   742   by (auto simp add: INF_def image_def)
```
```   743
```
```   744 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
```
```   745   by auto
```
```   746
```
```   747 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"
```
```   748   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
```
```   749   by (auto simp add: INF_def image_def)
```
```   750
```
```   751 lemma INT_cong [cong]:
```
```   752   "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)"
```
```   753   by (fact INF_cong)
```
```   754
```
```   755 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
```
```   756   by blast
```
```   757
```
```   758 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
```
```   759   by blast
```
```   760
```
```   761 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
```
```   762   by (fact INF_lower)
```
```   763
```
```   764 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
```
```   765   by (fact INF_greatest)
```
```   766
```
```   767 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
```
```   768   by (fact INF_empty)
```
```   769
```
```   770 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
```
```   771   by (fact INF_absorb)
```
```   772
```
```   773 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
```
```   774   by (fact le_INF_iff)
```
```   775
```
```   776 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
```
```   777   by (fact INF_insert)
```
```   778
```
```   779 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
```
```   780   by (fact INF_union)
```
```   781
```
```   782 lemma INT_insert_distrib:
```
```   783   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
```
```   784   by blast
```
```   785
```
```   786 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
```
```   787   by (fact INF_constant)
```
```   788
```
```   789 lemma INTER_UNIV_conv [simp]:
```
```   790  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
```
```   791  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
```
```   792   by (fact INF_top_conv)+
```
```   793
```
```   794 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
```
```   795   by (fact INF_UNIV_bool_expand)
```
```   796
```
```   797 lemma INT_anti_mono:
```
```   798   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
```
```   799   -- {* The last inclusion is POSITIVE! *}
```
```   800   by (fact INF_superset_mono)
```
```   801
```
```   802 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
```
```   803   by blast
```
```   804
```
```   805 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
```
```   806   by blast
```
```   807
```
```   808
```
```   809 subsection {* Union *}
```
```   810
```
```   811 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
```
```   812   "Union S \<equiv> \<Squnion>S"
```
```   813
```
```   814 notation (xsymbols)
```
```   815   Union  ("\<Union>_" [90] 90)
```
```   816
```
```   817 lemma Union_eq:
```
```   818   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
```
```   819 proof (rule set_eqI)
```
```   820   fix x
```
```   821   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
```
```   822     by auto
```
```   823   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
```
```   824     by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
```
```   825 qed
```
```   826
```
```   827 lemma Union_iff [simp, no_atp]:
```
```   828   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
```
```   829   by (unfold Union_eq) blast
```
```   830
```
```   831 lemma UnionI [intro]:
```
```   832   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
```
```   833   -- {* The order of the premises presupposes that @{term C} is rigid;
```
```   834     @{term A} may be flexible. *}
```
```   835   by auto
```
```   836
```
```   837 lemma UnionE [elim!]:
```
```   838   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
```
```   839   by auto
```
```   840
```
```   841 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
```
```   842   by (fact Sup_upper)
```
```   843
```
```   844 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
```
```   845   by (fact Sup_least)
```
```   846
```
```   847 lemma Union_empty [simp]: "\<Union>{} = {}"
```
```   848   by (fact Sup_empty)
```
```   849
```
```   850 lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
```
```   851   by (fact Sup_UNIV)
```
```   852
```
```   853 lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
```
```   854   by (fact Sup_insert)
```
```   855
```
```   856 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
```
```   857   by (fact Sup_union_distrib)
```
```   858
```
```   859 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
```
```   860   by (fact Sup_inter_less_eq)
```
```   861
```
```   862 lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```   863   by (fact Sup_bot_conv)
```
```   864
```
```   865 lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```   866   by (fact Sup_bot_conv)
```
```   867
```
```   868 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
```
```   869   by blast
```
```   870
```
```   871 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
```
```   872   by blast
```
```   873
```
```   874 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
```
```   875   by (fact Sup_subset_mono)
```
```   876
```
```   877
```
```   878 subsection {* Unions of families *}
```
```   879
```
```   880 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```   881   "UNION \<equiv> SUPR"
```
```   882
```
```   883 text {*
```
```   884   Note: must use name @{const UNION} here instead of @{text UN}
```
```   885   to allow the following syntax coexist with the plain constant name.
```
```   886 *}
```
```   887
```
```   888 syntax
```
```   889   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
```
```   890   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
```
```   891
```
```   892 syntax (xsymbols)
```
```   893   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
```
```   894   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
```
```   895
```
```   896 syntax (latex output)
```
```   897   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   898   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   899
```
```   900 translations
```
```   901   "UN x y. B"   == "UN x. UN y. B"
```
```   902   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
```
```   903   "UN x. B"     == "UN x:CONST UNIV. B"
```
```   904   "UN x:A. B"   == "CONST UNION A (%x. B)"
```
```   905
```
```   906 text {*
```
```   907   Note the difference between ordinary xsymbol syntax of indexed
```
```   908   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
```
```   909   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
```
```   910   former does not make the index expression a subscript of the
```
```   911   union/intersection symbol because this leads to problems with nested
```
```   912   subscripts in Proof General.
```
```   913 *}
```
```   914
```
```   915 print_translation {*
```
```   916   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
```
```   917 *} -- {* to avoid eta-contraction of body *}
```
```   918
```
```   919 lemma UNION_eq [no_atp]:
```
```   920   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
```
```   921   by (auto simp add: SUP_def)
```
```   922
```
```   923 lemma Union_image_eq [simp]:
```
```   924   "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
```
```   925   by (auto simp add: UNION_eq)
```
```   926
```
```   927 lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
```
```   928   by (auto simp add: SUP_def image_def)
```
```   929
```
```   930 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
```
```   931   -- {* The order of the premises presupposes that @{term A} is rigid;
```
```   932     @{term b} may be flexible. *}
```
```   933   by auto
```
```   934
```
```   935 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"
```
```   936   by (auto simp add: SUP_def image_def)
```
```   937
```
```   938 lemma UN_cong [cong]:
```
```   939   "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)"
```
```   940   by (fact SUP_cong)
```
```   941
```
```   942 lemma strong_UN_cong:
```
```   943   "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)"
```
```   944   by (unfold simp_implies_def) (fact UN_cong)
```
```   945
```
```   946 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
```
```   947   by blast
```
```   948
```
```   949 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
```
```   950   by (fact SUP_upper)
```
```   951
```
```   952 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
```
```   953   by (fact SUP_least)
```
```   954
```
```   955 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
```
```   956   by blast
```
```   957
```
```   958 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
```
```   959   by blast
```
```   960
```
```   961 lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}"
```
```   962   by (fact SUP_empty)
```
```   963
```
```   964 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
```
```   965   by (fact SUP_bot)
```
```   966
```
```   967 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
```
```   968   by (fact SUP_absorb)
```
```   969
```
```   970 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
```
```   971   by (fact SUP_insert)
```
```   972
```
```   973 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)"
```
```   974   by (fact SUP_union)
```
```   975
```
```   976 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)"
```
```   977   by blast
```
```   978
```
```   979 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
```
```   980   by (fact SUP_le_iff)
```
```   981
```
```   982 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
```
```   983   by (fact SUP_constant)
```
```   984
```
```   985 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
```
```   986   by blast
```
```   987
```
```   988 lemma UNION_empty_conv[simp]:
```
```   989   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```   990   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```   991   by (fact SUP_bot_conv)+
```
```   992
```
```   993 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
```
```   994   by blast
```
```   995
```
```   996 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
```
```   997   by blast
```
```   998
```
```   999 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
```
```  1000   by blast
```
```  1001
```
```  1002 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
```
```  1003   by (auto simp add: split_if_mem2)
```
```  1004
```
```  1005 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
```
```  1006   by (fact SUP_UNIV_bool_expand)
```
```  1007
```
```  1008 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
```
```  1009   by blast
```
```  1010
```
```  1011 lemma UN_mono:
```
```  1012   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
```
```  1013     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
```
```  1014   by (fact SUP_subset_mono)
```
```  1015
```
```  1016 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
```
```  1017   by blast
```
```  1018
```
```  1019 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
```
```  1020   by blast
```
```  1021
```
```  1022 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
```
```  1023   -- {* NOT suitable for rewriting *}
```
```  1024   by blast
```
```  1025
```
```  1026 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
```
```  1027   by blast
```
```  1028
```
```  1029
```
```  1030 subsection {* Distributive laws *}
```
```  1031
```
```  1032 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
```
```  1033   by (fact inf_Sup)
```
```  1034
```
```  1035 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
```
```  1036   by (fact sup_Inf)
```
```  1037
```
```  1038 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
```
```  1039   by (fact Sup_inf)
```
```  1040
```
```  1041 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)"
```
```  1042   by (rule sym) (rule INF_inf_distrib)
```
```  1043
```
```  1044 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)"
```
```  1045   by (rule sym) (rule SUP_sup_distrib)
```
```  1046
```
```  1047 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
```
```  1048   by (simp only: INT_Int_distrib INF_def)
```
```  1049
```
```  1050 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
```
```  1051   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
```
```  1052   -- {* Union of a family of unions *}
```
```  1053   by (simp only: UN_Un_distrib SUP_def)
```
```  1054
```
```  1055 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
```
```  1056   by (fact sup_INF)
```
```  1057
```
```  1058 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
```
```  1059   -- {* Halmos, Naive Set Theory, page 35. *}
```
```  1060   by (fact inf_SUP)
```
```  1061
```
```  1062 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)"
```
```  1063   by (fact SUP_inf_distrib2)
```
```  1064
```
```  1065 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)"
```
```  1066   by (fact INF_sup_distrib2)
```
```  1067
```
```  1068 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
```
```  1069   by (fact Sup_inf_eq_bot_iff)
```
```  1070
```
```  1071
```
```  1072 subsection {* Complement *}
```
```  1073
```
```  1074 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
```
```  1075   by (fact uminus_INF)
```
```  1076
```
```  1077 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
```
```  1078   by (fact uminus_SUP)
```
```  1079
```
```  1080
```
```  1081 subsection {* Miniscoping and maxiscoping *}
```
```  1082
```
```  1083 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
```
```  1084            and Intersections. *}
```
```  1085
```
```  1086 lemma UN_simps [simp]:
```
```  1087   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
```
```  1088   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
```
```  1089   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
```
```  1090   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
```
```  1091   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
```
```  1092   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
```
```  1093   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
```
```  1094   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
```
```  1095   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
```
```  1096   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
```
```  1097   by auto
```
```  1098
```
```  1099 lemma INT_simps [simp]:
```
```  1100   "\<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)"
```
```  1101   "\<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))"
```
```  1102   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
```
```  1103   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
```
```  1104   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
```
```  1105   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
```
```  1106   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
```
```  1107   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
```
```  1108   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
```
```  1109   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
```
```  1110   by auto
```
```  1111
```
```  1112 lemma UN_ball_bex_simps [simp, no_atp]:
```
```  1113   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
```
```  1114   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
```
```  1115   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
```
```  1116   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
```
```  1117   by auto
```
```  1118
```
```  1119
```
```  1120 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
```
```  1121
```
```  1122 lemma UN_extend_simps:
```
```  1123   "\<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)))"
```
```  1124   "\<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))"
```
```  1125   "\<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))"
```
```  1126   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
```
```  1127   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
```
```  1128   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
```
```  1129   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
```
```  1130   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
```
```  1131   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
```
```  1132   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
```
```  1133   by auto
```
```  1134
```
```  1135 lemma INT_extend_simps:
```
```  1136   "\<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))"
```
```  1137   "\<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))"
```
```  1138   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
```
```  1139   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
```
```  1140   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
```
```  1141   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
```
```  1142   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
```
```  1143   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
```
```  1144   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
```
```  1145   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
```
```  1146   by auto
```
```  1147
```
```  1148
```
```  1149 text {* Legacy names *}
```
```  1150
```
```  1151 lemma (in complete_lattice) Inf_singleton [simp]:
```
```  1152   "\<Sqinter>{a} = a"
```
```  1153   by (simp add: Inf_insert)
```
```  1154
```
```  1155 lemma (in complete_lattice) Sup_singleton [simp]:
```
```  1156   "\<Squnion>{a} = a"
```
```  1157   by (simp add: Sup_insert)
```
```  1158
```
```  1159 lemma (in complete_lattice) Inf_binary:
```
```  1160   "\<Sqinter>{a, b} = a \<sqinter> b"
```
```  1161   by (simp add: Inf_insert)
```
```  1162
```
```  1163 lemma (in complete_lattice) Sup_binary:
```
```  1164   "\<Squnion>{a, b} = a \<squnion> b"
```
```  1165   by (simp add: Sup_insert)
```
```  1166
```
```  1167 lemmas (in complete_lattice) INFI_def = INF_def
```
```  1168 lemmas (in complete_lattice) SUPR_def = SUP_def
```
```  1169 lemmas (in complete_lattice) INF_leI = INF_lower
```
```  1170 lemmas (in complete_lattice) INF_leI2 = INF_lower2
```
```  1171 lemmas (in complete_lattice) le_INFI = INF_greatest
```
```  1172 lemmas (in complete_lattice) le_SUPI = SUP_upper
```
```  1173 lemmas (in complete_lattice) le_SUPI2 = SUP_upper2
```
```  1174 lemmas (in complete_lattice) SUP_leI = SUP_least
```
```  1175 lemmas (in complete_lattice) less_INFD = less_INF_D
```
```  1176
```
```  1177 lemmas INFI_apply = INF_apply
```
```  1178 lemmas SUPR_apply = SUP_apply
```
```  1179
```
```  1180 text {* Grep and put to news from here *}
```
```  1181
```
```  1182 lemma (in complete_lattice) INF_eq:
```
```  1183   "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
```
```  1184   by (simp add: INF_def image_def)
```
```  1185
```
```  1186 lemma (in complete_lattice) SUP_eq:
```
```  1187   "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
```
```  1188   by (simp add: SUP_def image_def)
```
```  1189
```
```  1190 lemma (in complete_lattice) INF_subset:
```
```  1191   "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
```
```  1192   by (rule INF_superset_mono) auto
```
```  1193
```
```  1194 lemma (in complete_lattice) INF_UNIV_range:
```
```  1195   "(\<Sqinter>x. f x) = \<Sqinter>range f"
```
```  1196   by (fact INF_def)
```
```  1197
```
```  1198 lemma (in complete_lattice) SUP_UNIV_range:
```
```  1199   "(\<Squnion>x. f x) = \<Squnion>range f"
```
```  1200   by (fact SUP_def)
```
```  1201
```
```  1202 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
```
```  1203   by (simp add: Inf_insert)
```
```  1204
```
```  1205 lemma INTER_eq_Inter_image:
```
```  1206   "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
```
```  1207   by (fact INF_def)
```
```  1208
```
```  1209 lemma Inter_def:
```
```  1210   "\<Inter>S = (\<Inter>x\<in>S. x)"
```
```  1211   by (simp add: INTER_eq_Inter_image image_def)
```
```  1212
```
```  1213 lemmas INTER_def = INTER_eq
```
```  1214 lemmas UNION_def = UNION_eq
```
```  1215
```
```  1216 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
```
```  1217   by (fact INF_eq)
```
```  1218
```
```  1219 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
```
```  1220   by blast
```
```  1221
```
```  1222 lemma UNION_eq_Union_image:
```
```  1223   "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
```
```  1224   by (fact SUP_def)
```
```  1225
```
```  1226 lemma Union_def:
```
```  1227   "\<Union>S = (\<Union>x\<in>S. x)"
```
```  1228   by (simp add: UNION_eq_Union_image image_def)
```
```  1229
```
```  1230 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
```
```  1231   by blast
```
```  1232
```
```  1233 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
```
```  1234   by (fact SUP_eq)
```
```  1235
```
```  1236
```
```  1237 text {* Finally *}
```
```  1238
```
```  1239 no_notation
```
```  1240   less_eq  (infix "\<sqsubseteq>" 50) and
```
```  1241   less (infix "\<sqsubset>" 50) and
```
```  1242   bot ("\<bottom>") and
```
```  1243   top ("\<top>") and
```
```  1244   inf  (infixl "\<sqinter>" 70) and
```
```  1245   sup  (infixl "\<squnion>" 65) and
```
```  1246   Inf  ("\<Sqinter>_" [900] 900) and
```
```  1247   Sup  ("\<Squnion>_" [900] 900)
```
```  1248
```
```  1249 no_syntax (xsymbols)
```
```  1250   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
```
```  1251   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1252   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
```
```  1253   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1254
```
```  1255 lemmas mem_simps =
```
```  1256   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
```
```  1257   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
```
```  1258   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
```
```  1259
```
```  1260 end
```