src/HOL/Complete_Lattices.thy
 author wenzelm Fri Mar 07 22:30:58 2014 +0100 (2014-03-07) changeset 55990 41c6b99c5fb7 parent 54414 72949fae4f81 child 56015 57e2cfba9c6e permissions -rw-r--r--
more antiquotations;
```     1  (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Complete lattices *}
```
```     4
```
```     5 theory Complete_Lattices
```
```     6 imports Set
```
```     7 begin
```
```     8
```
```     9 notation
```
```    10   less_eq (infix "\<sqsubseteq>" 50) and
```
```    11   less (infix "\<sqsubset>" 50)
```
```    12
```
```    13
```
```    14 subsection {* Syntactic infimum and supremum operations *}
```
```    15
```
```    16 class Inf =
```
```    17   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```    18 begin
```
```    19
```
```    20 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    21   INF_def: "INFI A f = \<Sqinter>(f ` A)"
```
```    22
```
```    23 lemma INF_image [simp]: "INFI (f`A) g = INFI A (\<lambda>x. g (f x))"
```
```    24   by (simp add: INF_def image_image)
```
```    25
```
```    26 lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFI A C = INFI B D"
```
```    27   by (simp add: INF_def image_def)
```
```    28
```
```    29 end
```
```    30
```
```    31 class Sup =
```
```    32   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```    33 begin
```
```    34
```
```    35 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    36   SUP_def: "SUPR A f = \<Squnion>(f ` A)"
```
```    37
```
```    38 lemma SUP_image [simp]: "SUPR (f`A) g = SUPR A (%x. g (f x))"
```
```    39   by (simp add: SUP_def image_image)
```
```    40
```
```    41 lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPR A C = SUPR B D"
```
```    42   by (simp add: SUP_def image_def)
```
```    43
```
```    44 end
```
```    45
```
```    46 text {*
```
```    47   Note: must use names @{const INFI} and @{const SUPR} here instead of
```
```    48   @{text INF} and @{text SUP} to allow the following syntax coexist
```
```    49   with the plain constant names.
```
```    50 *}
```
```    51
```
```    52 syntax
```
```    53   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```    54   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
```
```    55   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```    56   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
```
```    57
```
```    58 syntax (xsymbols)
```
```    59   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
```
```    60   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
```
```    61   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
```
```    62   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
```
```    63
```
```    64 translations
```
```    65   "INF x y. B"   == "INF x. INF y. B"
```
```    66   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
```
```    67   "INF x. B"     == "INF x:CONST UNIV. B"
```
```    68   "INF x:A. B"   == "CONST INFI A (%x. B)"
```
```    69   "SUP x y. B"   == "SUP x. SUP y. B"
```
```    70   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
```
```    71   "SUP x. B"     == "SUP x:CONST UNIV. B"
```
```    72   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
```
```    73
```
```    74 print_translation {*
```
```    75   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
```
```    76     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
```
```    77 *} -- {* to avoid eta-contraction of body *}
```
```    78
```
```    79 subsection {* Abstract complete lattices *}
```
```    80
```
```    81 text {* A complete lattice always has a bottom and a top,
```
```    82 so we include them into the following type class,
```
```    83 along with assumptions that define bottom and top
```
```    84 in terms of infimum and supremum. *}
```
```    85
```
```    86 class complete_lattice = lattice + Inf + Sup + bot + top +
```
```    87   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```    88      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```    89   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```    90      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```    91   assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
```
```    92   assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
```
```    93 begin
```
```    94
```
```    95 subclass bounded_lattice
```
```    96 proof
```
```    97   fix a
```
```    98   show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
```
```    99   show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
```
```   100 qed
```
```   101
```
```   102 lemma dual_complete_lattice:
```
```   103   "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   104   by (auto intro!: class.complete_lattice.intro dual_lattice)
```
```   105     (unfold_locales, (fact Inf_empty Sup_empty
```
```   106         Sup_upper Sup_least Inf_lower Inf_greatest)+)
```
```   107
```
```   108 end
```
```   109
```
```   110 context complete_lattice
```
```   111 begin
```
```   112
```
```   113 lemma INF_foundation_dual:
```
```   114   "Sup.SUPR Inf = INFI"
```
```   115   by (simp add: fun_eq_iff INF_def Sup.SUP_def)
```
```   116
```
```   117 lemma SUP_foundation_dual:
```
```   118   "Inf.INFI Sup = SUPR" by (simp add: fun_eq_iff SUP_def Inf.INF_def)
```
```   119
```
```   120 lemma Sup_eqI:
```
```   121   "(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x"
```
```   122   by (blast intro: antisym Sup_least Sup_upper)
```
```   123
```
```   124 lemma Inf_eqI:
```
```   125   "(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x"
```
```   126   by (blast intro: antisym Inf_greatest Inf_lower)
```
```   127
```
```   128 lemma SUP_eqI:
```
```   129   "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x"
```
```   130   unfolding SUP_def by (rule Sup_eqI) auto
```
```   131
```
```   132 lemma INF_eqI:
```
```   133   "(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x"
```
```   134   unfolding INF_def by (rule Inf_eqI) auto
```
```   135
```
```   136 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
```
```   137   by (auto simp add: INF_def intro: Inf_lower)
```
```   138
```
```   139 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
```
```   140   by (auto simp add: INF_def intro: Inf_greatest)
```
```   141
```
```   142 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   143   by (auto simp add: SUP_def intro: Sup_upper)
```
```   144
```
```   145 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
```
```   146   by (auto simp add: SUP_def intro: Sup_least)
```
```   147
```
```   148 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
```
```   149   using Inf_lower [of u A] by auto
```
```   150
```
```   151 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
```
```   152   using INF_lower [of i A f] by auto
```
```   153
```
```   154 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
```
```   155   using Sup_upper [of u A] by auto
```
```   156
```
```   157 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   158   using SUP_upper [of i A f] by auto
```
```   159
```
```   160 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
```
```   161   by (auto intro: Inf_greatest dest: Inf_lower)
```
```   162
```
```   163 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
```
```   164   by (auto simp add: INF_def le_Inf_iff)
```
```   165
```
```   166 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
```
```   167   by (auto intro: Sup_least dest: Sup_upper)
```
```   168
```
```   169 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
```
```   170   by (auto simp add: SUP_def Sup_le_iff)
```
```   171
```
```   172 lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   173   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
```
```   174
```
```   175 lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
```
```   176   by (simp add: INF_def)
```
```   177
```
```   178 lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   179   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
```
```   180
```
```   181 lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
```
```   182   by (simp add: SUP_def)
```
```   183
```
```   184 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
```
```   185   by (simp add: INF_def)
```
```   186
```
```   187 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
```
```   188   by (simp add: SUP_def)
```
```   189
```
```   190 lemma Inf_UNIV [simp]:
```
```   191   "\<Sqinter>UNIV = \<bottom>"
```
```   192   by (auto intro!: antisym Inf_lower)
```
```   193
```
```   194 lemma Sup_UNIV [simp]:
```
```   195   "\<Squnion>UNIV = \<top>"
```
```   196   by (auto intro!: antisym Sup_upper)
```
```   197
```
```   198 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
```
```   199   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   200
```
```   201 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
```
```   202   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   203
```
```   204 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   205   by (auto intro: Inf_greatest Inf_lower)
```
```   206
```
```   207 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   208   by (auto intro: Sup_least Sup_upper)
```
```   209
```
```   210 lemma Inf_mono:
```
```   211   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
```
```   212   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   213 proof (rule Inf_greatest)
```
```   214   fix b assume "b \<in> B"
```
```   215   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
```
```   216   from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
```
```   217   with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
```
```   218 qed
```
```   219
```
```   220 lemma INF_mono:
```
```   221   "(\<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)"
```
```   222   unfolding INF_def by (rule Inf_mono) fast
```
```   223
```
```   224 lemma Sup_mono:
```
```   225   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
```
```   226   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   227 proof (rule Sup_least)
```
```   228   fix a assume "a \<in> A"
```
```   229   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
```
```   230   from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
```
```   231   with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
```
```   232 qed
```
```   233
```
```   234 lemma SUP_mono:
```
```   235   "(\<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)"
```
```   236   unfolding SUP_def by (rule Sup_mono) fast
```
```   237
```
```   238 lemma INF_superset_mono:
```
```   239   "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)"
```
```   240   -- {* The last inclusion is POSITIVE! *}
```
```   241   by (blast intro: INF_mono dest: subsetD)
```
```   242
```
```   243 lemma SUP_subset_mono:
```
```   244   "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)"
```
```   245   by (blast intro: SUP_mono dest: subsetD)
```
```   246
```
```   247 lemma Inf_less_eq:
```
```   248   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
```
```   249     and "A \<noteq> {}"
```
```   250   shows "\<Sqinter>A \<sqsubseteq> u"
```
```   251 proof -
```
```   252   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   253   moreover from `v \<in> A` assms(1) have "v \<sqsubseteq> u" by blast
```
```   254   ultimately show ?thesis by (rule Inf_lower2)
```
```   255 qed
```
```   256
```
```   257 lemma less_eq_Sup:
```
```   258   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
```
```   259     and "A \<noteq> {}"
```
```   260   shows "u \<sqsubseteq> \<Squnion>A"
```
```   261 proof -
```
```   262   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   263   moreover from `v \<in> A` assms(1) have "u \<sqsubseteq> v" by blast
```
```   264   ultimately show ?thesis by (rule Sup_upper2)
```
```   265 qed
```
```   266
```
```   267 lemma SUPR_eq:
```
```   268   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
```
```   269   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
```
```   270   shows "(SUP i:A. f i) = (SUP j:B. g j)"
```
```   271   by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
```
```   272
```
```   273 lemma INFI_eq:
```
```   274   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
```
```   275   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
```
```   276   shows "(INF i:A. f i) = (INF j:B. g j)"
```
```   277   by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
```
```   278
```
```   279 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
```
```   280   by (auto intro: Inf_greatest Inf_lower)
```
```   281
```
```   282 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
```
```   283   by (auto intro: Sup_least Sup_upper)
```
```   284
```
```   285 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
```
```   286   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
```
```   287
```
```   288 lemma INF_union:
```
```   289   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
```
```   290   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
```
```   291
```
```   292 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
```
```   293   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
```
```   294
```
```   295 lemma SUP_union:
```
```   296   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
```
```   297   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
```
```   298
```
```   299 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)"
```
```   300   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
```
```   301
```
```   302 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)" (is "?L = ?R")
```
```   303 proof (rule antisym)
```
```   304   show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
```
```   305 next
```
```   306   show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
```
```   307 qed
```
```   308
```
```   309 lemma Inf_top_conv [simp]:
```
```   310   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   311   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   312 proof -
```
```   313   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   314   proof
```
```   315     assume "\<forall>x\<in>A. x = \<top>"
```
```   316     then have "A = {} \<or> A = {\<top>}" by auto
```
```   317     then show "\<Sqinter>A = \<top>" by auto
```
```   318   next
```
```   319     assume "\<Sqinter>A = \<top>"
```
```   320     show "\<forall>x\<in>A. x = \<top>"
```
```   321     proof (rule ccontr)
```
```   322       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
```
```   323       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
```
```   324       then obtain B where "A = insert x B" by blast
```
```   325       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
```
```   326     qed
```
```   327   qed
```
```   328   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
```
```   329 qed
```
```   330
```
```   331 lemma INF_top_conv [simp]:
```
```   332  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   333  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   334   by (auto simp add: INF_def)
```
```   335
```
```   336 lemma Sup_bot_conv [simp]:
```
```   337   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
```
```   338   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
```
```   339   using dual_complete_lattice
```
```   340   by (rule complete_lattice.Inf_top_conv)+
```
```   341
```
```   342 lemma SUP_bot_conv [simp]:
```
```   343  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   344  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   345   by (auto simp add: SUP_def)
```
```   346
```
```   347 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
```
```   348   by (auto intro: antisym INF_lower INF_greatest)
```
```   349
```
```   350 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
```
```   351   by (auto intro: antisym SUP_upper SUP_least)
```
```   352
```
```   353 lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
```
```   354   by (cases "A = {}") simp_all
```
```   355
```
```   356 lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
```
```   357   by (cases "A = {}") simp_all
```
```   358
```
```   359 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)"
```
```   360   by (iprover intro: INF_lower INF_greatest order_trans antisym)
```
```   361
```
```   362 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)"
```
```   363   by (iprover intro: SUP_upper SUP_least order_trans antisym)
```
```   364
```
```   365 lemma INF_absorb:
```
```   366   assumes "k \<in> I"
```
```   367   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
```
```   368 proof -
```
```   369   from assms obtain J where "I = insert k J" by blast
```
```   370   then show ?thesis by (simp add: INF_insert)
```
```   371 qed
```
```   372
```
```   373 lemma SUP_absorb:
```
```   374   assumes "k \<in> I"
```
```   375   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
```
```   376 proof -
```
```   377   from assms obtain J where "I = insert k J" by blast
```
```   378   then show ?thesis by (simp add: SUP_insert)
```
```   379 qed
```
```   380
```
```   381 lemma INF_constant:
```
```   382   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
```
```   383   by simp
```
```   384
```
```   385 lemma SUP_constant:
```
```   386   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
```
```   387   by simp
```
```   388
```
```   389 lemma less_INF_D:
```
```   390   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
```
```   391 proof -
```
```   392   note `y < (\<Sqinter>i\<in>A. f i)`
```
```   393   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
```
```   394     by (rule INF_lower)
```
```   395   finally show "y < f i" .
```
```   396 qed
```
```   397
```
```   398 lemma SUP_lessD:
```
```   399   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
```
```   400 proof -
```
```   401   have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
```
```   402     by (rule SUP_upper)
```
```   403   also note `(\<Squnion>i\<in>A. f i) < y`
```
```   404   finally show "f i < y" .
```
```   405 qed
```
```   406
```
```   407 lemma INF_UNIV_bool_expand:
```
```   408   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
```
```   409   by (simp add: UNIV_bool INF_insert inf_commute)
```
```   410
```
```   411 lemma SUP_UNIV_bool_expand:
```
```   412   "(\<Squnion>b. A b) = A True \<squnion> A False"
```
```   413   by (simp add: UNIV_bool SUP_insert sup_commute)
```
```   414
```
```   415 lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
```
```   416   by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
```
```   417
```
```   418 lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f"
```
```   419   unfolding INF_def SUP_def by (rule Inf_le_Sup) auto
```
```   420
```
```   421 lemma SUP_eq_const:
```
```   422   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPR I f = x"
```
```   423   by (auto intro: SUP_eqI)
```
```   424
```
```   425 lemma INF_eq_const:
```
```   426   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFI I f = x"
```
```   427   by (auto intro: INF_eqI)
```
```   428
```
```   429 lemma SUP_eq_iff:
```
```   430   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> (SUPR I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   431   using SUP_eq_const[of I f c] SUP_upper[of _ I f] by (auto intro: antisym)
```
```   432
```
```   433 lemma INF_eq_iff:
```
```   434   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> (INFI I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   435   using INF_eq_const[of I f c] INF_lower[of _ I f] by (auto intro: antisym)
```
```   436
```
```   437 end
```
```   438
```
```   439 class complete_distrib_lattice = complete_lattice +
```
```   440   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
```
```   441   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   442 begin
```
```   443
```
```   444 lemma sup_INF:
```
```   445   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
```
```   446   by (simp add: INF_def sup_Inf image_image)
```
```   447
```
```   448 lemma inf_SUP:
```
```   449   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
```
```   450   by (simp add: SUP_def inf_Sup image_image)
```
```   451
```
```   452 lemma dual_complete_distrib_lattice:
```
```   453   "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   454   apply (rule class.complete_distrib_lattice.intro)
```
```   455   apply (fact dual_complete_lattice)
```
```   456   apply (rule class.complete_distrib_lattice_axioms.intro)
```
```   457   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
```
```   458   done
```
```   459
```
```   460 subclass distrib_lattice proof
```
```   461   fix a b c
```
```   462   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
```
```   463   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
```
```   464 qed
```
```   465
```
```   466 lemma Inf_sup:
```
```   467   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
```
```   468   by (simp add: sup_Inf sup_commute)
```
```   469
```
```   470 lemma Sup_inf:
```
```   471   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
```
```   472   by (simp add: inf_Sup inf_commute)
```
```   473
```
```   474 lemma INF_sup:
```
```   475   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
```
```   476   by (simp add: sup_INF sup_commute)
```
```   477
```
```   478 lemma SUP_inf:
```
```   479   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
```
```   480   by (simp add: inf_SUP inf_commute)
```
```   481
```
```   482 lemma Inf_sup_eq_top_iff:
```
```   483   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
```
```   484   by (simp only: Inf_sup INF_top_conv)
```
```   485
```
```   486 lemma Sup_inf_eq_bot_iff:
```
```   487   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
```
```   488   by (simp only: Sup_inf SUP_bot_conv)
```
```   489
```
```   490 lemma INF_sup_distrib2:
```
```   491   "(\<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)"
```
```   492   by (subst INF_commute) (simp add: sup_INF INF_sup)
```
```   493
```
```   494 lemma SUP_inf_distrib2:
```
```   495   "(\<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)"
```
```   496   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
```
```   497
```
```   498 end
```
```   499
```
```   500 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
```
```   501 begin
```
```   502
```
```   503 lemma dual_complete_boolean_algebra:
```
```   504   "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
```
```   505   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
```
```   506
```
```   507 lemma uminus_Inf:
```
```   508   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
```
```   509 proof (rule antisym)
```
```   510   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
```
```   511     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
```
```   512   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
```
```   513     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
```
```   514 qed
```
```   515
```
```   516 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
```
```   517   by (simp add: INF_def SUP_def uminus_Inf image_image)
```
```   518
```
```   519 lemma uminus_Sup:
```
```   520   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
```
```   521 proof -
```
```   522   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
```
```   523   then show ?thesis by simp
```
```   524 qed
```
```   525
```
```   526 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
```
```   527   by (simp add: INF_def SUP_def uminus_Sup image_image)
```
```   528
```
```   529 end
```
```   530
```
```   531 class complete_linorder = linorder + complete_lattice
```
```   532 begin
```
```   533
```
```   534 lemma dual_complete_linorder:
```
```   535   "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   536   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
```
```   537
```
```   538 lemma complete_linorder_inf_min: "inf = min"
```
```   539   by (auto intro: antisym simp add: min_def fun_eq_iff)
```
```   540
```
```   541 lemma complete_linorder_sup_max: "sup = max"
```
```   542   by (auto intro: antisym simp add: max_def fun_eq_iff)
```
```   543
```
```   544 lemma Inf_less_iff:
```
```   545   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
```
```   546   unfolding not_le [symmetric] le_Inf_iff by auto
```
```   547
```
```   548 lemma INF_less_iff:
```
```   549   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
```
```   550   unfolding INF_def Inf_less_iff by auto
```
```   551
```
```   552 lemma less_Sup_iff:
```
```   553   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
```
```   554   unfolding not_le [symmetric] Sup_le_iff by auto
```
```   555
```
```   556 lemma less_SUP_iff:
```
```   557   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
```
```   558   unfolding SUP_def less_Sup_iff by auto
```
```   559
```
```   560 lemma Sup_eq_top_iff [simp]:
```
```   561   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
```
```   562 proof
```
```   563   assume *: "\<Squnion>A = \<top>"
```
```   564   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
```
```   565   proof (intro allI impI)
```
```   566     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
```
```   567       unfolding less_Sup_iff by auto
```
```   568   qed
```
```   569 next
```
```   570   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
```
```   571   show "\<Squnion>A = \<top>"
```
```   572   proof (rule ccontr)
```
```   573     assume "\<Squnion>A \<noteq> \<top>"
```
```   574     with top_greatest [of "\<Squnion>A"]
```
```   575     have "\<Squnion>A < \<top>" unfolding le_less by auto
```
```   576     then have "\<Squnion>A < \<Squnion>A"
```
```   577       using * unfolding less_Sup_iff by auto
```
```   578     then show False by auto
```
```   579   qed
```
```   580 qed
```
```   581
```
```   582 lemma SUP_eq_top_iff [simp]:
```
```   583   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
```
```   584   unfolding SUP_def by auto
```
```   585
```
```   586 lemma Inf_eq_bot_iff [simp]:
```
```   587   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
```
```   588   using dual_complete_linorder
```
```   589   by (rule complete_linorder.Sup_eq_top_iff)
```
```   590
```
```   591 lemma INF_eq_bot_iff [simp]:
```
```   592   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
```
```   593   unfolding INF_def by auto
```
```   594
```
```   595 lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
```
```   596 proof safe
```
```   597   fix y assume "x \<le> \<Squnion>A" "y < x"
```
```   598   then have "y < \<Squnion>A" by auto
```
```   599   then show "\<exists>a\<in>A. y < a"
```
```   600     unfolding less_Sup_iff .
```
```   601 qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
```
```   602
```
```   603 lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
```
```   604   unfolding le_Sup_iff SUP_def by simp
```
```   605
```
```   606 lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
```
```   607 proof safe
```
```   608   fix y assume "x \<ge> \<Sqinter>A" "y > x"
```
```   609   then have "y > \<Sqinter>A" by auto
```
```   610   then show "\<exists>a\<in>A. y > a"
```
```   611     unfolding Inf_less_iff .
```
```   612 qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
```
```   613
```
```   614 lemma INF_le_iff:
```
```   615   "INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
```
```   616   unfolding Inf_le_iff INF_def by simp
```
```   617
```
```   618 subclass complete_distrib_lattice
```
```   619 proof
```
```   620   fix a and B
```
```   621   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   622     by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
```
```   623       (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
```
```   624         le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
```
```   625 qed
```
```   626
```
```   627 end
```
```   628
```
```   629
```
```   630 subsection {* Complete lattice on @{typ bool} *}
```
```   631
```
```   632 instantiation bool :: complete_lattice
```
```   633 begin
```
```   634
```
```   635 definition
```
```   636   [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
```
```   637
```
```   638 definition
```
```   639   [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
```
```   640
```
```   641 instance proof
```
```   642 qed (auto intro: bool_induct)
```
```   643
```
```   644 end
```
```   645
```
```   646 lemma not_False_in_image_Ball [simp]:
```
```   647   "False \<notin> P ` A \<longleftrightarrow> Ball A P"
```
```   648   by auto
```
```   649
```
```   650 lemma True_in_image_Bex [simp]:
```
```   651   "True \<in> P ` A \<longleftrightarrow> Bex A P"
```
```   652   by auto
```
```   653
```
```   654 lemma INF_bool_eq [simp]:
```
```   655   "INFI = Ball"
```
```   656   by (simp add: fun_eq_iff INF_def)
```
```   657
```
```   658 lemma SUP_bool_eq [simp]:
```
```   659   "SUPR = Bex"
```
```   660   by (simp add: fun_eq_iff SUP_def)
```
```   661
```
```   662 instance bool :: complete_boolean_algebra proof
```
```   663 qed (auto intro: bool_induct)
```
```   664
```
```   665
```
```   666 subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
```
```   667
```
```   668 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   669 begin
```
```   670
```
```   671 definition
```
```   672   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
```
```   673
```
```   674 lemma Inf_apply [simp, code]:
```
```   675   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
```
```   676   by (simp add: Inf_fun_def)
```
```   677
```
```   678 definition
```
```   679   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
```
```   680
```
```   681 lemma Sup_apply [simp, code]:
```
```   682   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
```
```   683   by (simp add: Sup_fun_def)
```
```   684
```
```   685 instance proof
```
```   686 qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
```
```   687
```
```   688 end
```
```   689
```
```   690 lemma INF_apply [simp]:
```
```   691   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
```
```   692   by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def)
```
```   693
```
```   694 lemma SUP_apply [simp]:
```
```   695   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
```
```   696   by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def)
```
```   697
```
```   698 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
```
```   699 qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image)
```
```   700
```
```   701 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
```
```   702
```
```   703
```
```   704 subsection {* Complete lattice on unary and binary predicates *}
```
```   705
```
```   706 lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
```
```   707   by simp
```
```   708
```
```   709 lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
```
```   710   by simp
```
```   711
```
```   712 lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
```
```   713   by auto
```
```   714
```
```   715 lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
```
```   716   by auto
```
```   717
```
```   718 lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
```
```   719   by auto
```
```   720
```
```   721 lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
```
```   722   by auto
```
```   723
```
```   724 lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
```
```   725   by auto
```
```   726
```
```   727 lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
```
```   728   by auto
```
```   729
```
```   730 lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
```
```   731   by simp
```
```   732
```
```   733 lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
```
```   734   by simp
```
```   735
```
```   736 lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
```
```   737   by auto
```
```   738
```
```   739 lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
```
```   740   by auto
```
```   741
```
```   742 lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
```
```   743   by auto
```
```   744
```
```   745 lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
```
```   746   by auto
```
```   747
```
```   748
```
```   749 subsection {* Complete lattice on @{typ "_ set"} *}
```
```   750
```
```   751 instantiation "set" :: (type) complete_lattice
```
```   752 begin
```
```   753
```
```   754 definition
```
```   755   "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
```
```   756
```
```   757 definition
```
```   758   "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
```
```   759
```
```   760 instance proof
```
```   761 qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
```
```   762
```
```   763 end
```
```   764
```
```   765 instance "set" :: (type) complete_boolean_algebra
```
```   766 proof
```
```   767 qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
```
```   768
```
```   769
```
```   770 subsubsection {* Inter *}
```
```   771
```
```   772 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
```
```   773   "Inter S \<equiv> \<Sqinter>S"
```
```   774
```
```   775 notation (xsymbols)
```
```   776   Inter  ("\<Inter>_" [900] 900)
```
```   777
```
```   778 lemma Inter_eq:
```
```   779   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
```
```   780 proof (rule set_eqI)
```
```   781   fix x
```
```   782   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
```
```   783     by auto
```
```   784   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
```
```   785     by (simp add: Inf_set_def image_def)
```
```   786 qed
```
```   787
```
```   788 lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
```
```   789   by (unfold Inter_eq) blast
```
```   790
```
```   791 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
```
```   792   by (simp add: Inter_eq)
```
```   793
```
```   794 text {*
```
```   795   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
```
```   796   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
```
```   797   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
```
```   798 *}
```
```   799
```
```   800 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
```
```   801   by auto
```
```   802
```
```   803 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
```
```   804   -- {* ``Classical'' elimination rule -- does not require proving
```
```   805     @{prop "X \<in> C"}. *}
```
```   806   by (unfold Inter_eq) blast
```
```   807
```
```   808 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   809   by (fact Inf_lower)
```
```   810
```
```   811 lemma Inter_subset:
```
```   812   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   813   by (fact Inf_less_eq)
```
```   814
```
```   815 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
```
```   816   by (fact Inf_greatest)
```
```   817
```
```   818 lemma Inter_empty: "\<Inter>{} = UNIV"
```
```   819   by (fact Inf_empty) (* already simp *)
```
```   820
```
```   821 lemma Inter_UNIV: "\<Inter>UNIV = {}"
```
```   822   by (fact Inf_UNIV) (* already simp *)
```
```   823
```
```   824 lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
```
```   825   by (fact Inf_insert) (* already simp *)
```
```   826
```
```   827 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
```
```   828   by (fact less_eq_Inf_inter)
```
```   829
```
```   830 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
```
```   831   by (fact Inf_union_distrib)
```
```   832
```
```   833 lemma Inter_UNIV_conv [simp]:
```
```   834   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   835   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   836   by (fact Inf_top_conv)+
```
```   837
```
```   838 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
```
```   839   by (fact Inf_superset_mono)
```
```   840
```
```   841
```
```   842 subsubsection {* Intersections of families *}
```
```   843
```
```   844 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```   845   "INTER \<equiv> INFI"
```
```   846
```
```   847 text {*
```
```   848   Note: must use name @{const INTER} here instead of @{text INT}
```
```   849   to allow the following syntax coexist with the plain constant name.
```
```   850 *}
```
```   851
```
```   852 syntax
```
```   853   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
```
```   854   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
```
```   855
```
```   856 syntax (xsymbols)
```
```   857   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
```
```   858   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
```
```   859
```
```   860 syntax (latex output)
```
```   861   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   862   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   863
```
```   864 translations
```
```   865   "INT x y. B"  == "INT x. INT y. B"
```
```   866   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
```
```   867   "INT x. B"    == "INT x:CONST UNIV. B"
```
```   868   "INT x:A. B"  == "CONST INTER A (%x. B)"
```
```   869
```
```   870 print_translation {*
```
```   871   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
```
```   872 *} -- {* to avoid eta-contraction of body *}
```
```   873
```
```   874 lemma INTER_eq:
```
```   875   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
```
```   876   by (auto simp add: INF_def)
```
```   877
```
```   878 lemma Inter_image_eq [simp]:
```
```   879   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
```
```   880   by (rule sym) (fact INF_def)
```
```   881
```
```   882 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
```
```   883   by (auto simp add: INF_def image_def)
```
```   884
```
```   885 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
```
```   886   by (auto simp add: INF_def image_def)
```
```   887
```
```   888 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
```
```   889   by auto
```
```   890
```
```   891 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"
```
```   892   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
```
```   893   by (auto simp add: INF_def image_def)
```
```   894
```
```   895 lemma INT_cong [cong]:
```
```   896   "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)"
```
```   897   by (fact INF_cong)
```
```   898
```
```   899 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
```
```   900   by blast
```
```   901
```
```   902 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
```
```   903   by blast
```
```   904
```
```   905 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
```
```   906   by (fact INF_lower)
```
```   907
```
```   908 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
```
```   909   by (fact INF_greatest)
```
```   910
```
```   911 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
```
```   912   by (fact INF_empty)
```
```   913
```
```   914 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
```
```   915   by (fact INF_absorb)
```
```   916
```
```   917 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
```
```   918   by (fact le_INF_iff)
```
```   919
```
```   920 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
```
```   921   by (fact INF_insert)
```
```   922
```
```   923 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
```
```   924   by (fact INF_union)
```
```   925
```
```   926 lemma INT_insert_distrib:
```
```   927   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
```
```   928   by blast
```
```   929
```
```   930 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
```
```   931   by (fact INF_constant)
```
```   932
```
```   933 lemma INTER_UNIV_conv:
```
```   934  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
```
```   935  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
```
```   936   by (fact INF_top_conv)+ (* already simp *)
```
```   937
```
```   938 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
```
```   939   by (fact INF_UNIV_bool_expand)
```
```   940
```
```   941 lemma INT_anti_mono:
```
```   942   "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)"
```
```   943   -- {* The last inclusion is POSITIVE! *}
```
```   944   by (fact INF_superset_mono)
```
```   945
```
```   946 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
```
```   947   by blast
```
```   948
```
```   949 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
```
```   950   by blast
```
```   951
```
```   952
```
```   953 subsubsection {* Union *}
```
```   954
```
```   955 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
```
```   956   "Union S \<equiv> \<Squnion>S"
```
```   957
```
```   958 notation (xsymbols)
```
```   959   Union  ("\<Union>_" [900] 900)
```
```   960
```
```   961 lemma Union_eq:
```
```   962   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
```
```   963 proof (rule set_eqI)
```
```   964   fix x
```
```   965   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
```
```   966     by auto
```
```   967   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
```
```   968     by (simp add: Sup_set_def image_def)
```
```   969 qed
```
```   970
```
```   971 lemma Union_iff [simp]:
```
```   972   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
```
```   973   by (unfold Union_eq) blast
```
```   974
```
```   975 lemma UnionI [intro]:
```
```   976   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
```
```   977   -- {* The order of the premises presupposes that @{term C} is rigid;
```
```   978     @{term A} may be flexible. *}
```
```   979   by auto
```
```   980
```
```   981 lemma UnionE [elim!]:
```
```   982   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
```
```   983   by auto
```
```   984
```
```   985 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
```
```   986   by (fact Sup_upper)
```
```   987
```
```   988 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
```
```   989   by (fact Sup_least)
```
```   990
```
```   991 lemma Union_empty: "\<Union>{} = {}"
```
```   992   by (fact Sup_empty) (* already simp *)
```
```   993
```
```   994 lemma Union_UNIV: "\<Union>UNIV = UNIV"
```
```   995   by (fact Sup_UNIV) (* already simp *)
```
```   996
```
```   997 lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
```
```   998   by (fact Sup_insert) (* already simp *)
```
```   999
```
```  1000 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
```
```  1001   by (fact Sup_union_distrib)
```
```  1002
```
```  1003 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
```
```  1004   by (fact Sup_inter_less_eq)
```
```  1005
```
```  1006 lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1007   by (fact Sup_bot_conv) (* already simp *)
```
```  1008
```
```  1009 lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1010   by (fact Sup_bot_conv) (* already simp *)
```
```  1011
```
```  1012 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
```
```  1013   by blast
```
```  1014
```
```  1015 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
```
```  1016   by blast
```
```  1017
```
```  1018 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
```
```  1019   by (fact Sup_subset_mono)
```
```  1020
```
```  1021
```
```  1022 subsubsection {* Unions of families *}
```
```  1023
```
```  1024 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```  1025   "UNION \<equiv> SUPR"
```
```  1026
```
```  1027 text {*
```
```  1028   Note: must use name @{const UNION} here instead of @{text UN}
```
```  1029   to allow the following syntax coexist with the plain constant name.
```
```  1030 *}
```
```  1031
```
```  1032 syntax
```
```  1033   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
```
```  1034   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
```
```  1035
```
```  1036 syntax (xsymbols)
```
```  1037   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
```
```  1038   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1039
```
```  1040 syntax (latex output)
```
```  1041   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```  1042   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```  1043
```
```  1044 translations
```
```  1045   "UN x y. B"   == "UN x. UN y. B"
```
```  1046   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
```
```  1047   "UN x. B"     == "UN x:CONST UNIV. B"
```
```  1048   "UN x:A. B"   == "CONST UNION A (%x. B)"
```
```  1049
```
```  1050 text {*
```
```  1051   Note the difference between ordinary xsymbol syntax of indexed
```
```  1052   unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
```
```  1053   and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
```
```  1054   former does not make the index expression a subscript of the
```
```  1055   union/intersection symbol because this leads to problems with nested
```
```  1056   subscripts in Proof General.
```
```  1057 *}
```
```  1058
```
```  1059 print_translation {*
```
```  1060   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
```
```  1061 *} -- {* to avoid eta-contraction of body *}
```
```  1062
```
```  1063 lemma UNION_eq:
```
```  1064   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
```
```  1065   by (auto simp add: SUP_def)
```
```  1066
```
```  1067 lemma bind_UNION [code]:
```
```  1068   "Set.bind A f = UNION A f"
```
```  1069   by (simp add: bind_def UNION_eq)
```
```  1070
```
```  1071 lemma member_bind [simp]:
```
```  1072   "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
```
```  1073   by (simp add: bind_UNION)
```
```  1074
```
```  1075 lemma Union_image_eq [simp]:
```
```  1076   "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
```
```  1077   by (rule sym) (fact SUP_def)
```
```  1078
```
```  1079 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
```
```  1080   by (auto simp add: SUP_def image_def)
```
```  1081
```
```  1082 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
```
```  1083   -- {* The order of the premises presupposes that @{term A} is rigid;
```
```  1084     @{term b} may be flexible. *}
```
```  1085   by auto
```
```  1086
```
```  1087 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"
```
```  1088   by (auto simp add: SUP_def image_def)
```
```  1089
```
```  1090 lemma UN_cong [cong]:
```
```  1091   "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)"
```
```  1092   by (fact SUP_cong)
```
```  1093
```
```  1094 lemma strong_UN_cong:
```
```  1095   "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)"
```
```  1096   by (unfold simp_implies_def) (fact UN_cong)
```
```  1097
```
```  1098 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
```
```  1099   by blast
```
```  1100
```
```  1101 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
```
```  1102   by (fact SUP_upper)
```
```  1103
```
```  1104 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
```
```  1105   by (fact SUP_least)
```
```  1106
```
```  1107 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
```
```  1108   by blast
```
```  1109
```
```  1110 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
```
```  1111   by blast
```
```  1112
```
```  1113 lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
```
```  1114   by (fact SUP_empty)
```
```  1115
```
```  1116 lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
```
```  1117   by (fact SUP_bot) (* already simp *)
```
```  1118
```
```  1119 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
```
```  1120   by (fact SUP_absorb)
```
```  1121
```
```  1122 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
```
```  1123   by (fact SUP_insert)
```
```  1124
```
```  1125 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)"
```
```  1126   by (fact SUP_union)
```
```  1127
```
```  1128 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)"
```
```  1129   by blast
```
```  1130
```
```  1131 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
```
```  1132   by (fact SUP_le_iff)
```
```  1133
```
```  1134 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
```
```  1135   by (fact SUP_constant)
```
```  1136
```
```  1137 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
```
```  1138   by blast
```
```  1139
```
```  1140 lemma UNION_empty_conv:
```
```  1141   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1142   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1143   by (fact SUP_bot_conv)+ (* already simp *)
```
```  1144
```
```  1145 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
```
```  1146   by blast
```
```  1147
```
```  1148 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
```
```  1149   by blast
```
```  1150
```
```  1151 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
```
```  1152   by blast
```
```  1153
```
```  1154 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
```
```  1155   by (auto simp add: split_if_mem2)
```
```  1156
```
```  1157 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
```
```  1158   by (fact SUP_UNIV_bool_expand)
```
```  1159
```
```  1160 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
```
```  1161   by blast
```
```  1162
```
```  1163 lemma UN_mono:
```
```  1164   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
```
```  1165     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
```
```  1166   by (fact SUP_subset_mono)
```
```  1167
```
```  1168 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
```
```  1169   by blast
```
```  1170
```
```  1171 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
```
```  1172   by blast
```
```  1173
```
```  1174 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
```
```  1175   -- {* NOT suitable for rewriting *}
```
```  1176   by blast
```
```  1177
```
```  1178 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
```
```  1179   by blast
```
```  1180
```
```  1181 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
```
```  1182   by blast
```
```  1183
```
```  1184
```
```  1185 subsubsection {* Distributive laws *}
```
```  1186
```
```  1187 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
```
```  1188   by (fact inf_Sup)
```
```  1189
```
```  1190 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
```
```  1191   by (fact sup_Inf)
```
```  1192
```
```  1193 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
```
```  1194   by (fact Sup_inf)
```
```  1195
```
```  1196 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)"
```
```  1197   by (rule sym) (rule INF_inf_distrib)
```
```  1198
```
```  1199 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)"
```
```  1200   by (rule sym) (rule SUP_sup_distrib)
```
```  1201
```
```  1202 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
```
```  1203   by (simp only: INT_Int_distrib INF_def)
```
```  1204
```
```  1205 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
```
```  1206   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
```
```  1207   -- {* Union of a family of unions *}
```
```  1208   by (simp only: UN_Un_distrib SUP_def)
```
```  1209
```
```  1210 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
```
```  1211   by (fact sup_INF)
```
```  1212
```
```  1213 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
```
```  1214   -- {* Halmos, Naive Set Theory, page 35. *}
```
```  1215   by (fact inf_SUP)
```
```  1216
```
```  1217 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)"
```
```  1218   by (fact SUP_inf_distrib2)
```
```  1219
```
```  1220 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)"
```
```  1221   by (fact INF_sup_distrib2)
```
```  1222
```
```  1223 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
```
```  1224   by (fact Sup_inf_eq_bot_iff)
```
```  1225
```
```  1226
```
```  1227 subsubsection {* Complement *}
```
```  1228
```
```  1229 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
```
```  1230   by (fact uminus_INF)
```
```  1231
```
```  1232 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
```
```  1233   by (fact uminus_SUP)
```
```  1234
```
```  1235
```
```  1236 subsubsection {* Miniscoping and maxiscoping *}
```
```  1237
```
```  1238 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
```
```  1239            and Intersections. *}
```
```  1240
```
```  1241 lemma UN_simps [simp]:
```
```  1242   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
```
```  1243   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
```
```  1244   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
```
```  1245   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
```
```  1246   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
```
```  1247   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
```
```  1248   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
```
```  1249   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
```
```  1250   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
```
```  1251   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
```
```  1252   by auto
```
```  1253
```
```  1254 lemma INT_simps [simp]:
```
```  1255   "\<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)"
```
```  1256   "\<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))"
```
```  1257   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
```
```  1258   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
```
```  1259   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
```
```  1260   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
```
```  1261   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
```
```  1262   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
```
```  1263   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
```
```  1264   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
```
```  1265   by auto
```
```  1266
```
```  1267 lemma UN_ball_bex_simps [simp]:
```
```  1268   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
```
```  1269   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
```
```  1270   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
```
```  1271   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
```
```  1272   by auto
```
```  1273
```
```  1274
```
```  1275 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
```
```  1276
```
```  1277 lemma UN_extend_simps:
```
```  1278   "\<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)))"
```
```  1279   "\<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))"
```
```  1280   "\<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))"
```
```  1281   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
```
```  1282   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
```
```  1283   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
```
```  1284   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
```
```  1285   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
```
```  1286   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
```
```  1287   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
```
```  1288   by auto
```
```  1289
```
```  1290 lemma INT_extend_simps:
```
```  1291   "\<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))"
```
```  1292   "\<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))"
```
```  1293   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
```
```  1294   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
```
```  1295   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
```
```  1296   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
```
```  1297   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
```
```  1298   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
```
```  1299   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
```
```  1300   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
```
```  1301   by auto
```
```  1302
```
```  1303 text {* Finally *}
```
```  1304
```
```  1305 no_notation
```
```  1306   less_eq (infix "\<sqsubseteq>" 50) and
```
```  1307   less (infix "\<sqsubset>" 50)
```
```  1308
```
```  1309 lemmas mem_simps =
```
```  1310   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
```
```  1311   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
```
```  1312   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
```
```  1313
```
```  1314 end
```
```  1315
```