src/HOL/Complete_Lattices.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63099 af0e964aad7b child 63172 d4f459eb7ed0 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 section \<open>Complete lattices\<close>
```
```     4
```
```     5 theory Complete_Lattices
```
```     6 imports Fun
```
```     7 begin
```
```     8
```
```     9 notation
```
```    10   less_eq (infix "\<sqsubseteq>" 50) and
```
```    11   less (infix "\<sqsubset>" 50)
```
```    12
```
```    13
```
```    14 subsection \<open>Syntactic infimum and supremum operations\<close>
```
```    15
```
```    16 class Inf =
```
```    17   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```    18 begin
```
```    19
```
```    20 abbreviation INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```    21 where
```
```    22   "INFIMUM A f \<equiv> \<Sqinter>(f ` A)"
```
```    23
```
```    24 lemma INF_image [simp]:
```
```    25   "INFIMUM (f ` A) g = INFIMUM A (g \<circ> f)"
```
```    26   by (simp add: image_comp)
```
```    27
```
```    28 lemma INF_identity_eq [simp]:
```
```    29   "INFIMUM A (\<lambda>x. x) = \<Sqinter>A"
```
```    30   by simp
```
```    31
```
```    32 lemma INF_id_eq [simp]:
```
```    33   "INFIMUM A id = \<Sqinter>A"
```
```    34   by simp
```
```    35
```
```    36 lemma INF_cong:
```
```    37   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
```
```    38   by (simp add: image_def)
```
```    39
```
```    40 lemma strong_INF_cong [cong]:
```
```    41   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
```
```    42   unfolding simp_implies_def by (fact INF_cong)
```
```    43
```
```    44 end
```
```    45
```
```    46 class Sup =
```
```    47   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```    48 begin
```
```    49
```
```    50 abbreviation SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```    51 where
```
```    52   "SUPREMUM A f \<equiv> \<Squnion>(f ` A)"
```
```    53
```
```    54 lemma SUP_image [simp]:
```
```    55   "SUPREMUM (f ` A) g = SUPREMUM A (g \<circ> f)"
```
```    56   by (simp add: image_comp)
```
```    57
```
```    58 lemma SUP_identity_eq [simp]:
```
```    59   "SUPREMUM A (\<lambda>x. x) = \<Squnion>A"
```
```    60   by simp
```
```    61
```
```    62 lemma SUP_id_eq [simp]:
```
```    63   "SUPREMUM A id = \<Squnion>A"
```
```    64   by (simp add: id_def)
```
```    65
```
```    66 lemma SUP_cong:
```
```    67   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
```
```    68   by (simp add: image_def)
```
```    69
```
```    70 lemma strong_SUP_cong [cong]:
```
```    71   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
```
```    72   unfolding simp_implies_def by (fact SUP_cong)
```
```    73
```
```    74 end
```
```    75
```
```    76 text \<open>
```
```    77   Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of
```
```    78   \<open>INF\<close> and \<open>SUP\<close> to allow the following syntax coexist
```
```    79   with the plain constant names.
```
```    80 \<close>
```
```    81
```
```    82 syntax (ASCII)
```
```    83   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```    84   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
```
```    85   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```    86   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
```
```    87
```
```    88 syntax (output)
```
```    89   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```    90   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
```
```    91   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```    92   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
```
```    93
```
```    94 syntax
```
```    95   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
```
```    96   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
```
```    97   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
```
```    98   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
```
```    99
```
```   100 translations
```
```   101   "\<Sqinter>x y. B"   \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. B"
```
```   102   "\<Sqinter>x. B"     \<rightleftharpoons> "CONST INFIMUM CONST UNIV (\<lambda>x. B)"
```
```   103   "\<Sqinter>x. B"     \<rightleftharpoons> "\<Sqinter>x \<in> CONST UNIV. B"
```
```   104   "\<Sqinter>x\<in>A. B"   \<rightleftharpoons> "CONST INFIMUM A (\<lambda>x. B)"
```
```   105   "\<Squnion>x y. B"   \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. B"
```
```   106   "\<Squnion>x. B"     \<rightleftharpoons> "CONST SUPREMUM CONST UNIV (\<lambda>x. B)"
```
```   107   "\<Squnion>x. B"     \<rightleftharpoons> "\<Squnion>x \<in> CONST UNIV. B"
```
```   108   "\<Squnion>x\<in>A. B"   \<rightleftharpoons> "CONST SUPREMUM A (\<lambda>x. B)"
```
```   109
```
```   110 print_translation \<open>
```
```   111   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"},
```
```   112     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}]
```
```   113 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
```
```   114
```
```   115
```
```   116 subsection \<open>Abstract complete lattices\<close>
```
```   117
```
```   118 text \<open>A complete lattice always has a bottom and a top,
```
```   119 so we include them into the following type class,
```
```   120 along with assumptions that define bottom and top
```
```   121 in terms of infimum and supremum.\<close>
```
```   122
```
```   123 class complete_lattice = lattice + Inf + Sup + bot + top +
```
```   124   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```   125      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```   126   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```   127      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```   128   assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
```
```   129   assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
```
```   130 begin
```
```   131
```
```   132 subclass bounded_lattice
```
```   133 proof
```
```   134   fix a
```
```   135   show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
```
```   136   show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
```
```   137 qed
```
```   138
```
```   139 lemma dual_complete_lattice:
```
```   140   "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   141   by (auto intro!: class.complete_lattice.intro dual_lattice)
```
```   142     (unfold_locales, (fact Inf_empty Sup_empty
```
```   143         Sup_upper Sup_least Inf_lower Inf_greatest)+)
```
```   144
```
```   145 end
```
```   146
```
```   147 context complete_lattice
```
```   148 begin
```
```   149
```
```   150 lemma Sup_eqI:
```
```   151   "(\<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"
```
```   152   by (blast intro: antisym Sup_least Sup_upper)
```
```   153
```
```   154 lemma Inf_eqI:
```
```   155   "(\<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"
```
```   156   by (blast intro: antisym Inf_greatest Inf_lower)
```
```   157
```
```   158 lemma SUP_eqI:
```
```   159   "(\<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"
```
```   160   using Sup_eqI [of "f ` A" x] by auto
```
```   161
```
```   162 lemma INF_eqI:
```
```   163   "(\<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"
```
```   164   using Inf_eqI [of "f ` A" x] by auto
```
```   165
```
```   166 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
```
```   167   using Inf_lower [of _ "f ` A"] by simp
```
```   168
```
```   169 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
```
```   170   using Inf_greatest [of "f ` A"] by auto
```
```   171
```
```   172 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   173   using Sup_upper [of _ "f ` A"] by simp
```
```   174
```
```   175 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
```
```   176   using Sup_least [of "f ` A"] by auto
```
```   177
```
```   178 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
```
```   179   using Inf_lower [of u A] by auto
```
```   180
```
```   181 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
```
```   182   using INF_lower [of i A f] by auto
```
```   183
```
```   184 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
```
```   185   using Sup_upper [of u A] by auto
```
```   186
```
```   187 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   188   using SUP_upper [of i A f] by auto
```
```   189
```
```   190 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
```
```   191   by (auto intro: Inf_greatest dest: Inf_lower)
```
```   192
```
```   193 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
```
```   194   using le_Inf_iff [of _ "f ` A"] by simp
```
```   195
```
```   196 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
```
```   197   by (auto intro: Sup_least dest: Sup_upper)
```
```   198
```
```   199 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
```
```   200   using Sup_le_iff [of "f ` A"] by simp
```
```   201
```
```   202 lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   203   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
```
```   204
```
```   205 lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFIMUM A f"
```
```   206   by (simp cong del: strong_INF_cong)
```
```   207
```
```   208 lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   209   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
```
```   210
```
```   211 lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPREMUM A f"
```
```   212   by (simp cong del: strong_SUP_cong)
```
```   213
```
```   214 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
```
```   215   by (simp cong del: strong_INF_cong)
```
```   216
```
```   217 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
```
```   218   by (simp cong del: strong_SUP_cong)
```
```   219
```
```   220 lemma Inf_UNIV [simp]:
```
```   221   "\<Sqinter>UNIV = \<bottom>"
```
```   222   by (auto intro!: antisym Inf_lower)
```
```   223
```
```   224 lemma Sup_UNIV [simp]:
```
```   225   "\<Squnion>UNIV = \<top>"
```
```   226   by (auto intro!: antisym Sup_upper)
```
```   227
```
```   228 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
```
```   229   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   230
```
```   231 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
```
```   232   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   233
```
```   234 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   235   by (auto intro: Inf_greatest Inf_lower)
```
```   236
```
```   237 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   238   by (auto intro: Sup_least Sup_upper)
```
```   239
```
```   240 lemma Inf_mono:
```
```   241   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
```
```   242   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   243 proof (rule Inf_greatest)
```
```   244   fix b assume "b \<in> B"
```
```   245   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
```
```   246   from \<open>a \<in> A\<close> have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
```
```   247   with \<open>a \<sqsubseteq> b\<close> show "\<Sqinter>A \<sqsubseteq> b" by auto
```
```   248 qed
```
```   249
```
```   250 lemma INF_mono:
```
```   251   "(\<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)"
```
```   252   using Inf_mono [of "g ` B" "f ` A"] by auto
```
```   253
```
```   254 lemma Sup_mono:
```
```   255   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
```
```   256   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   257 proof (rule Sup_least)
```
```   258   fix a assume "a \<in> A"
```
```   259   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
```
```   260   from \<open>b \<in> B\<close> have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
```
```   261   with \<open>a \<sqsubseteq> b\<close> show "a \<sqsubseteq> \<Squnion>B" by auto
```
```   262 qed
```
```   263
```
```   264 lemma SUP_mono:
```
```   265   "(\<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)"
```
```   266   using Sup_mono [of "f ` A" "g ` B"] by auto
```
```   267
```
```   268 lemma INF_superset_mono:
```
```   269   "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)"
```
```   270   \<comment> \<open>The last inclusion is POSITIVE!\<close>
```
```   271   by (blast intro: INF_mono dest: subsetD)
```
```   272
```
```   273 lemma SUP_subset_mono:
```
```   274   "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)"
```
```   275   by (blast intro: SUP_mono dest: subsetD)
```
```   276
```
```   277 lemma Inf_less_eq:
```
```   278   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
```
```   279     and "A \<noteq> {}"
```
```   280   shows "\<Sqinter>A \<sqsubseteq> u"
```
```   281 proof -
```
```   282   from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
```
```   283   moreover from \<open>v \<in> A\<close> assms(1) have "v \<sqsubseteq> u" by blast
```
```   284   ultimately show ?thesis by (rule Inf_lower2)
```
```   285 qed
```
```   286
```
```   287 lemma less_eq_Sup:
```
```   288   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
```
```   289     and "A \<noteq> {}"
```
```   290   shows "u \<sqsubseteq> \<Squnion>A"
```
```   291 proof -
```
```   292   from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
```
```   293   moreover from \<open>v \<in> A\<close> assms(1) have "u \<sqsubseteq> v" by blast
```
```   294   ultimately show ?thesis by (rule Sup_upper2)
```
```   295 qed
```
```   296
```
```   297 lemma INF_eq:
```
```   298   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
```
```   299   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
```
```   300   shows "INFIMUM A f = INFIMUM B g"
```
```   301   by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
```
```   302
```
```   303 lemma SUP_eq:
```
```   304   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
```
```   305   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
```
```   306   shows "SUPREMUM A f = SUPREMUM B g"
```
```   307   by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
```
```   308
```
```   309 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
```
```   310   by (auto intro: Inf_greatest Inf_lower)
```
```   311
```
```   312 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
```
```   313   by (auto intro: Sup_least Sup_upper)
```
```   314
```
```   315 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
```
```   316   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
```
```   317
```
```   318 lemma INF_union:
```
```   319   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
```
```   320   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
```
```   321
```
```   322 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
```
```   323   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
```
```   324
```
```   325 lemma SUP_union:
```
```   326   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
```
```   327   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
```
```   328
```
```   329 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)"
```
```   330   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
```
```   331
```
```   332 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")
```
```   333 proof (rule antisym)
```
```   334   show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
```
```   335 next
```
```   336   show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
```
```   337 qed
```
```   338
```
```   339 lemma Inf_top_conv [simp]:
```
```   340   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   341   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   342 proof -
```
```   343   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   344   proof
```
```   345     assume "\<forall>x\<in>A. x = \<top>"
```
```   346     then have "A = {} \<or> A = {\<top>}" by auto
```
```   347     then show "\<Sqinter>A = \<top>" by auto
```
```   348   next
```
```   349     assume "\<Sqinter>A = \<top>"
```
```   350     show "\<forall>x\<in>A. x = \<top>"
```
```   351     proof (rule ccontr)
```
```   352       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
```
```   353       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
```
```   354       then obtain B where "A = insert x B" by blast
```
```   355       with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp
```
```   356     qed
```
```   357   qed
```
```   358   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
```
```   359 qed
```
```   360
```
```   361 lemma INF_top_conv [simp]:
```
```   362   "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   363   "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   364   using Inf_top_conv [of "B ` A"] by simp_all
```
```   365
```
```   366 lemma Sup_bot_conv [simp]:
```
```   367   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
```
```   368   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
```
```   369   using dual_complete_lattice
```
```   370   by (rule complete_lattice.Inf_top_conv)+
```
```   371
```
```   372 lemma SUP_bot_conv [simp]:
```
```   373  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   374  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   375   using Sup_bot_conv [of "B ` A"] by simp_all
```
```   376
```
```   377 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
```
```   378   by (auto intro: antisym INF_lower INF_greatest)
```
```   379
```
```   380 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
```
```   381   by (auto intro: antisym SUP_upper SUP_least)
```
```   382
```
```   383 lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
```
```   384   by (cases "A = {}") simp_all
```
```   385
```
```   386 lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
```
```   387   by (cases "A = {}") simp_all
```
```   388
```
```   389 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)"
```
```   390   by (iprover intro: INF_lower INF_greatest order_trans antisym)
```
```   391
```
```   392 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)"
```
```   393   by (iprover intro: SUP_upper SUP_least order_trans antisym)
```
```   394
```
```   395 lemma INF_absorb:
```
```   396   assumes "k \<in> I"
```
```   397   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
```
```   398 proof -
```
```   399   from assms obtain J where "I = insert k J" by blast
```
```   400   then show ?thesis by simp
```
```   401 qed
```
```   402
```
```   403 lemma SUP_absorb:
```
```   404   assumes "k \<in> I"
```
```   405   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
```
```   406 proof -
```
```   407   from assms obtain J where "I = insert k J" by blast
```
```   408   then show ?thesis by simp
```
```   409 qed
```
```   410
```
```   411 lemma INF_inf_const1:
```
```   412   "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
```
```   413   by (intro antisym INF_greatest inf_mono order_refl INF_lower)
```
```   414      (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
```
```   415
```
```   416 lemma INF_inf_const2:
```
```   417   "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
```
```   418   using INF_inf_const1[of I x f] by (simp add: inf_commute)
```
```   419
```
```   420 lemma INF_constant:
```
```   421   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
```
```   422   by simp
```
```   423
```
```   424 lemma SUP_constant:
```
```   425   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
```
```   426   by simp
```
```   427
```
```   428 lemma less_INF_D:
```
```   429   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
```
```   430 proof -
```
```   431   note \<open>y < (\<Sqinter>i\<in>A. f i)\<close>
```
```   432   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close>
```
```   433     by (rule INF_lower)
```
```   434   finally show "y < f i" .
```
```   435 qed
```
```   436
```
```   437 lemma SUP_lessD:
```
```   438   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
```
```   439 proof -
```
```   440   have "f i \<le> (\<Squnion>i\<in>A. f i)" using \<open>i \<in> A\<close>
```
```   441     by (rule SUP_upper)
```
```   442   also note \<open>(\<Squnion>i\<in>A. f i) < y\<close>
```
```   443   finally show "f i < y" .
```
```   444 qed
```
```   445
```
```   446 lemma INF_UNIV_bool_expand:
```
```   447   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
```
```   448   by (simp add: UNIV_bool inf_commute)
```
```   449
```
```   450 lemma SUP_UNIV_bool_expand:
```
```   451   "(\<Squnion>b. A b) = A True \<squnion> A False"
```
```   452   by (simp add: UNIV_bool sup_commute)
```
```   453
```
```   454 lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
```
```   455   by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
```
```   456
```
```   457 lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
```
```   458   using Inf_le_Sup [of "f ` A"] by simp
```
```   459
```
```   460 lemma INF_eq_const:
```
```   461   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
```
```   462   by (auto intro: INF_eqI)
```
```   463
```
```   464 lemma SUP_eq_const:
```
```   465   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
```
```   466   by (auto intro: SUP_eqI)
```
```   467
```
```   468 lemma INF_eq_iff:
```
```   469   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> (INFIMUM I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   470   using INF_eq_const [of I f c] INF_lower [of _ I f]
```
```   471   by (auto intro: antisym cong del: strong_INF_cong)
```
```   472
```
```   473 lemma SUP_eq_iff:
```
```   474   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> (SUPREMUM I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   475   using SUP_eq_const [of I f c] SUP_upper [of _ I f]
```
```   476   by (auto intro: antisym cong del: strong_SUP_cong)
```
```   477
```
```   478 end
```
```   479
```
```   480 class complete_distrib_lattice = complete_lattice +
```
```   481   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
```
```   482   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   483 begin
```
```   484
```
```   485 lemma sup_INF:
```
```   486   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
```
```   487   unfolding sup_Inf by simp
```
```   488
```
```   489 lemma inf_SUP:
```
```   490   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
```
```   491   unfolding inf_Sup by simp
```
```   492
```
```   493 lemma dual_complete_distrib_lattice:
```
```   494   "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   495   apply (rule class.complete_distrib_lattice.intro)
```
```   496   apply (fact dual_complete_lattice)
```
```   497   apply (rule class.complete_distrib_lattice_axioms.intro)
```
```   498   apply (simp_all add: inf_Sup sup_Inf)
```
```   499   done
```
```   500
```
```   501 subclass distrib_lattice proof
```
```   502   fix a b c
```
```   503   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
```
```   504   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by simp
```
```   505 qed
```
```   506
```
```   507 lemma Inf_sup:
```
```   508   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
```
```   509   by (simp add: sup_Inf sup_commute)
```
```   510
```
```   511 lemma Sup_inf:
```
```   512   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
```
```   513   by (simp add: inf_Sup inf_commute)
```
```   514
```
```   515 lemma INF_sup:
```
```   516   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
```
```   517   by (simp add: sup_INF sup_commute)
```
```   518
```
```   519 lemma SUP_inf:
```
```   520   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
```
```   521   by (simp add: inf_SUP inf_commute)
```
```   522
```
```   523 lemma Inf_sup_eq_top_iff:
```
```   524   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
```
```   525   by (simp only: Inf_sup INF_top_conv)
```
```   526
```
```   527 lemma Sup_inf_eq_bot_iff:
```
```   528   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
```
```   529   by (simp only: Sup_inf SUP_bot_conv)
```
```   530
```
```   531 lemma INF_sup_distrib2:
```
```   532   "(\<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)"
```
```   533   by (subst INF_commute) (simp add: sup_INF INF_sup)
```
```   534
```
```   535 lemma SUP_inf_distrib2:
```
```   536   "(\<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)"
```
```   537   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
```
```   538
```
```   539 context
```
```   540   fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
```
```   541   assumes "mono f"
```
```   542 begin
```
```   543
```
```   544 lemma mono_Inf:
```
```   545   shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
```
```   546   using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
```
```   547
```
```   548 lemma mono_Sup:
```
```   549   shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
```
```   550   using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
```
```   551
```
```   552 lemma mono_INF:
```
```   553   "f (INF i : I. A i) \<le> (INF x : I. f (A x))"
```
```   554   by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower)
```
```   555
```
```   556 lemma mono_SUP:
```
```   557   "(SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
```
```   558   by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper)
```
```   559
```
```   560 end
```
```   561
```
```   562 end
```
```   563
```
```   564 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
```
```   565 begin
```
```   566
```
```   567 lemma dual_complete_boolean_algebra:
```
```   568   "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
```
```   569   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
```
```   570
```
```   571 lemma uminus_Inf:
```
```   572   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
```
```   573 proof (rule antisym)
```
```   574   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
```
```   575     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
```
```   576   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
```
```   577     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
```
```   578 qed
```
```   579
```
```   580 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
```
```   581   by (simp add: uminus_Inf image_image)
```
```   582
```
```   583 lemma uminus_Sup:
```
```   584   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
```
```   585 proof -
```
```   586   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
```
```   587   then show ?thesis by simp
```
```   588 qed
```
```   589
```
```   590 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
```
```   591   by (simp add: uminus_Sup image_image)
```
```   592
```
```   593 end
```
```   594
```
```   595 class complete_linorder = linorder + complete_lattice
```
```   596 begin
```
```   597
```
```   598 lemma dual_complete_linorder:
```
```   599   "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   600   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
```
```   601
```
```   602 lemma complete_linorder_inf_min: "inf = min"
```
```   603   by (auto intro: antisym simp add: min_def fun_eq_iff)
```
```   604
```
```   605 lemma complete_linorder_sup_max: "sup = max"
```
```   606   by (auto intro: antisym simp add: max_def fun_eq_iff)
```
```   607
```
```   608 lemma Inf_less_iff:
```
```   609   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
```
```   610   unfolding not_le [symmetric] le_Inf_iff by auto
```
```   611
```
```   612 lemma INF_less_iff:
```
```   613   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
```
```   614   using Inf_less_iff [of "f ` A"] by simp
```
```   615
```
```   616 lemma less_Sup_iff:
```
```   617   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
```
```   618   unfolding not_le [symmetric] Sup_le_iff by auto
```
```   619
```
```   620 lemma less_SUP_iff:
```
```   621   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
```
```   622   using less_Sup_iff [of _ "f ` A"] by simp
```
```   623
```
```   624 lemma Sup_eq_top_iff [simp]:
```
```   625   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
```
```   626 proof
```
```   627   assume *: "\<Squnion>A = \<top>"
```
```   628   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
```
```   629   proof (intro allI impI)
```
```   630     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
```
```   631       unfolding less_Sup_iff by auto
```
```   632   qed
```
```   633 next
```
```   634   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
```
```   635   show "\<Squnion>A = \<top>"
```
```   636   proof (rule ccontr)
```
```   637     assume "\<Squnion>A \<noteq> \<top>"
```
```   638     with top_greatest [of "\<Squnion>A"]
```
```   639     have "\<Squnion>A < \<top>" unfolding le_less by auto
```
```   640     then have "\<Squnion>A < \<Squnion>A"
```
```   641       using * unfolding less_Sup_iff by auto
```
```   642     then show False by auto
```
```   643   qed
```
```   644 qed
```
```   645
```
```   646 lemma SUP_eq_top_iff [simp]:
```
```   647   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
```
```   648   using Sup_eq_top_iff [of "f ` A"] by simp
```
```   649
```
```   650 lemma Inf_eq_bot_iff [simp]:
```
```   651   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
```
```   652   using dual_complete_linorder
```
```   653   by (rule complete_linorder.Sup_eq_top_iff)
```
```   654
```
```   655 lemma INF_eq_bot_iff [simp]:
```
```   656   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
```
```   657   using Inf_eq_bot_iff [of "f ` A"] by simp
```
```   658
```
```   659 lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
```
```   660 proof safe
```
```   661   fix y assume "x \<ge> \<Sqinter>A" "y > x"
```
```   662   then have "y > \<Sqinter>A" by auto
```
```   663   then show "\<exists>a\<in>A. y > a"
```
```   664     unfolding Inf_less_iff .
```
```   665 qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
```
```   666
```
```   667 lemma INF_le_iff:
```
```   668   "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
```
```   669   using Inf_le_iff [of "f ` A"] by simp
```
```   670
```
```   671 lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
```
```   672 proof safe
```
```   673   fix y assume "x \<le> \<Squnion>A" "y < x"
```
```   674   then have "y < \<Squnion>A" by auto
```
```   675   then show "\<exists>a\<in>A. y < a"
```
```   676     unfolding less_Sup_iff .
```
```   677 qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
```
```   678
```
```   679 lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
```
```   680   using le_Sup_iff [of _ "f ` A"] by simp
```
```   681
```
```   682 subclass complete_distrib_lattice
```
```   683 proof
```
```   684   fix a and B
```
```   685   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   686     by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
```
```   687       (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
```
```   688         le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
```
```   689 qed
```
```   690
```
```   691 end
```
```   692
```
```   693
```
```   694 subsection \<open>Complete lattice on @{typ bool}\<close>
```
```   695
```
```   696 instantiation bool :: complete_lattice
```
```   697 begin
```
```   698
```
```   699 definition
```
```   700   [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
```
```   701
```
```   702 definition
```
```   703   [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
```
```   704
```
```   705 instance proof
```
```   706 qed (auto intro: bool_induct)
```
```   707
```
```   708 end
```
```   709
```
```   710 lemma not_False_in_image_Ball [simp]:
```
```   711   "False \<notin> P ` A \<longleftrightarrow> Ball A P"
```
```   712   by auto
```
```   713
```
```   714 lemma True_in_image_Bex [simp]:
```
```   715   "True \<in> P ` A \<longleftrightarrow> Bex A P"
```
```   716   by auto
```
```   717
```
```   718 lemma INF_bool_eq [simp]:
```
```   719   "INFIMUM = Ball"
```
```   720   by (simp add: fun_eq_iff)
```
```   721
```
```   722 lemma SUP_bool_eq [simp]:
```
```   723   "SUPREMUM = Bex"
```
```   724   by (simp add: fun_eq_iff)
```
```   725
```
```   726 instance bool :: complete_boolean_algebra proof
```
```   727 qed (auto intro: bool_induct)
```
```   728
```
```   729
```
```   730 subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<close>
```
```   731
```
```   732 instantiation "fun" :: (type, Inf) Inf
```
```   733 begin
```
```   734
```
```   735 definition
```
```   736   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
```
```   737
```
```   738 lemma Inf_apply [simp, code]:
```
```   739   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
```
```   740   by (simp add: Inf_fun_def)
```
```   741
```
```   742 instance ..
```
```   743
```
```   744 end
```
```   745
```
```   746 instantiation "fun" :: (type, Sup) Sup
```
```   747 begin
```
```   748
```
```   749 definition
```
```   750   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
```
```   751
```
```   752 lemma Sup_apply [simp, code]:
```
```   753   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
```
```   754   by (simp add: Sup_fun_def)
```
```   755
```
```   756 instance ..
```
```   757
```
```   758 end
```
```   759
```
```   760 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   761 begin
```
```   762
```
```   763 instance proof
```
```   764 qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
```
```   765
```
```   766 end
```
```   767
```
```   768 lemma INF_apply [simp]:
```
```   769   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
```
```   770   using Inf_apply [of "f ` A"] by (simp add: comp_def)
```
```   771
```
```   772 lemma SUP_apply [simp]:
```
```   773   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
```
```   774   using Sup_apply [of "f ` A"] by (simp add: comp_def)
```
```   775
```
```   776 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
```
```   777 qed (auto simp add: inf_Sup sup_Inf fun_eq_iff image_image)
```
```   778
```
```   779 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
```
```   780
```
```   781
```
```   782 subsection \<open>Complete lattice on unary and binary predicates\<close>
```
```   783
```
```   784 lemma Inf1_I:
```
```   785   "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
```
```   786   by auto
```
```   787
```
```   788 lemma INF1_I:
```
```   789   "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
```
```   790   by simp
```
```   791
```
```   792 lemma INF2_I:
```
```   793   "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
```
```   794   by simp
```
```   795
```
```   796 lemma Inf2_I:
```
```   797   "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
```
```   798   by auto
```
```   799
```
```   800 lemma Inf1_D:
```
```   801   "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
```
```   802   by auto
```
```   803
```
```   804 lemma INF1_D:
```
```   805   "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
```
```   806   by simp
```
```   807
```
```   808 lemma Inf2_D:
```
```   809   "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
```
```   810   by auto
```
```   811
```
```   812 lemma INF2_D:
```
```   813   "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
```
```   814   by simp
```
```   815
```
```   816 lemma Inf1_E:
```
```   817   assumes "(\<Sqinter>A) a"
```
```   818   obtains "P a" | "P \<notin> A"
```
```   819   using assms by auto
```
```   820
```
```   821 lemma INF1_E:
```
```   822   assumes "(\<Sqinter>x\<in>A. B x) b"
```
```   823   obtains "B a b" | "a \<notin> A"
```
```   824   using assms by auto
```
```   825
```
```   826 lemma Inf2_E:
```
```   827   assumes "(\<Sqinter>A) a b"
```
```   828   obtains "r a b" | "r \<notin> A"
```
```   829   using assms by auto
```
```   830
```
```   831 lemma INF2_E:
```
```   832   assumes "(\<Sqinter>x\<in>A. B x) b c"
```
```   833   obtains "B a b c" | "a \<notin> A"
```
```   834   using assms by auto
```
```   835
```
```   836 lemma Sup1_I:
```
```   837   "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
```
```   838   by auto
```
```   839
```
```   840 lemma SUP1_I:
```
```   841   "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
```
```   842   by auto
```
```   843
```
```   844 lemma Sup2_I:
```
```   845   "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
```
```   846   by auto
```
```   847
```
```   848 lemma SUP2_I:
```
```   849   "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
```
```   850   by auto
```
```   851
```
```   852 lemma Sup1_E:
```
```   853   assumes "(\<Squnion>A) a"
```
```   854   obtains P where "P \<in> A" and "P a"
```
```   855   using assms by auto
```
```   856
```
```   857 lemma SUP1_E:
```
```   858   assumes "(\<Squnion>x\<in>A. B x) b"
```
```   859   obtains x where "x \<in> A" and "B x b"
```
```   860   using assms by auto
```
```   861
```
```   862 lemma Sup2_E:
```
```   863   assumes "(\<Squnion>A) a b"
```
```   864   obtains r where "r \<in> A" "r a b"
```
```   865   using assms by auto
```
```   866
```
```   867 lemma SUP2_E:
```
```   868   assumes "(\<Squnion>x\<in>A. B x) b c"
```
```   869   obtains x where "x \<in> A" "B x b c"
```
```   870   using assms by auto
```
```   871
```
```   872
```
```   873 subsection \<open>Complete lattice on @{typ "_ set"}\<close>
```
```   874
```
```   875 instantiation "set" :: (type) complete_lattice
```
```   876 begin
```
```   877
```
```   878 definition
```
```   879   "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
```
```   880
```
```   881 definition
```
```   882   "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
```
```   883
```
```   884 instance proof
```
```   885 qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
```
```   886
```
```   887 end
```
```   888
```
```   889 instance "set" :: (type) complete_boolean_algebra
```
```   890 proof
```
```   891 qed (auto simp add: Inf_set_def Sup_set_def image_def)
```
```   892
```
```   893
```
```   894 subsubsection \<open>Inter\<close>
```
```   895
```
```   896 abbreviation Inter :: "'a set set \<Rightarrow> 'a set"  ("\<Inter>_" [900] 900)
```
```   897   where "\<Inter>S \<equiv> \<Sqinter>S"
```
```   898
```
```   899 lemma Inter_eq:
```
```   900   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
```
```   901 proof (rule set_eqI)
```
```   902   fix x
```
```   903   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
```
```   904     by auto
```
```   905   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
```
```   906     by (simp add: Inf_set_def image_def)
```
```   907 qed
```
```   908
```
```   909 lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
```
```   910   by (unfold Inter_eq) blast
```
```   911
```
```   912 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
```
```   913   by (simp add: Inter_eq)
```
```   914
```
```   915 text \<open>
```
```   916   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
```
```   917   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
```
```   918   @{prop "X \<in> C"} does not!  This rule is analogous to \<open>spec\<close>.
```
```   919 \<close>
```
```   920
```
```   921 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
```
```   922   by auto
```
```   923
```
```   924 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
```
```   925   \<comment> \<open>``Classical'' elimination rule -- does not require proving
```
```   926     @{prop "X \<in> C"}.\<close>
```
```   927   by (unfold Inter_eq) blast
```
```   928
```
```   929 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   930   by (fact Inf_lower)
```
```   931
```
```   932 lemma Inter_subset:
```
```   933   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   934   by (fact Inf_less_eq)
```
```   935
```
```   936 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A"
```
```   937   by (fact Inf_greatest)
```
```   938
```
```   939 lemma Inter_empty: "\<Inter>{} = UNIV"
```
```   940   by (fact Inf_empty) (* already simp *)
```
```   941
```
```   942 lemma Inter_UNIV: "\<Inter>UNIV = {}"
```
```   943   by (fact Inf_UNIV) (* already simp *)
```
```   944
```
```   945 lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
```
```   946   by (fact Inf_insert) (* already simp *)
```
```   947
```
```   948 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
```
```   949   by (fact less_eq_Inf_inter)
```
```   950
```
```   951 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
```
```   952   by (fact Inf_union_distrib)
```
```   953
```
```   954 lemma Inter_UNIV_conv [simp]:
```
```   955   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   956   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   957   by (fact Inf_top_conv)+
```
```   958
```
```   959 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
```
```   960   by (fact Inf_superset_mono)
```
```   961
```
```   962
```
```   963 subsubsection \<open>Intersections of families\<close>
```
```   964
```
```   965 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
```
```   966   where "INTER \<equiv> INFIMUM"
```
```   967
```
```   968 text \<open>
```
```   969   Note: must use name @{const INTER} here instead of \<open>INT\<close>
```
```   970   to allow the following syntax coexist with the plain constant name.
```
```   971 \<close>
```
```   972
```
```   973 syntax (ASCII)
```
```   974   "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3INT _./ _)" [0, 10] 10)
```
```   975   "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
```
```   976
```
```   977 syntax (latex output)
```
```   978   "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   979   "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   980
```
```   981 syntax
```
```   982   "_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
```
```   983   "_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
```
```   984
```
```   985 translations
```
```   986   "\<Inter>x y. B"  \<rightleftharpoons> "\<Inter>x. \<Inter>y. B"
```
```   987   "\<Inter>x. B"    \<rightleftharpoons> "CONST INTER CONST UNIV (\<lambda>x. B)"
```
```   988   "\<Inter>x. B"    \<rightleftharpoons> "\<Inter>x \<in> CONST UNIV. B"
```
```   989   "\<Inter>x\<in>A. B"  \<rightleftharpoons> "CONST INTER A (\<lambda>x. B)"
```
```   990
```
```   991 print_translation \<open>
```
```   992   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
```
```   993 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
```
```   994
```
```   995 lemma INTER_eq:
```
```   996   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
```
```   997   by (auto intro!: INF_eqI)
```
```   998
```
```   999 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
```
```  1000   using Inter_iff [of _ "B ` A"] by simp
```
```  1001
```
```  1002 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
```
```  1003   by auto
```
```  1004
```
```  1005 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
```
```  1006   by auto
```
```  1007
```
```  1008 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"
```
```  1009   \<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close>
```
```  1010   by auto
```
```  1011
```
```  1012 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
```
```  1013   by blast
```
```  1014
```
```  1015 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
```
```  1016   by blast
```
```  1017
```
```  1018 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
```
```  1019   by (fact INF_lower)
```
```  1020
```
```  1021 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
```
```  1022   by (fact INF_greatest)
```
```  1023
```
```  1024 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
```
```  1025   by (fact INF_empty)
```
```  1026
```
```  1027 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
```
```  1028   by (fact INF_absorb)
```
```  1029
```
```  1030 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
```
```  1031   by (fact le_INF_iff)
```
```  1032
```
```  1033 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
```
```  1034   by (fact INF_insert)
```
```  1035
```
```  1036 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
```
```  1037   by (fact INF_union)
```
```  1038
```
```  1039 lemma INT_insert_distrib:
```
```  1040   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
```
```  1041   by blast
```
```  1042
```
```  1043 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
```
```  1044   by (fact INF_constant)
```
```  1045
```
```  1046 lemma INTER_UNIV_conv:
```
```  1047  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
```
```  1048  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
```
```  1049   by (fact INF_top_conv)+ (* already simp *)
```
```  1050
```
```  1051 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
```
```  1052   by (fact INF_UNIV_bool_expand)
```
```  1053
```
```  1054 lemma INT_anti_mono:
```
```  1055   "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)"
```
```  1056   \<comment> \<open>The last inclusion is POSITIVE!\<close>
```
```  1057   by (fact INF_superset_mono)
```
```  1058
```
```  1059 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
```
```  1060   by blast
```
```  1061
```
```  1062 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
```
```  1063   by blast
```
```  1064
```
```  1065
```
```  1066 subsubsection \<open>Union\<close>
```
```  1067
```
```  1068 abbreviation Union :: "'a set set \<Rightarrow> 'a set"  ("\<Union>_" [900] 900)
```
```  1069   where "\<Union>S \<equiv> \<Squnion>S"
```
```  1070
```
```  1071 lemma Union_eq:
```
```  1072   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
```
```  1073 proof (rule set_eqI)
```
```  1074   fix x
```
```  1075   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
```
```  1076     by auto
```
```  1077   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
```
```  1078     by (simp add: Sup_set_def image_def)
```
```  1079 qed
```
```  1080
```
```  1081 lemma Union_iff [simp]:
```
```  1082   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
```
```  1083   by (unfold Union_eq) blast
```
```  1084
```
```  1085 lemma UnionI [intro]:
```
```  1086   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
```
```  1087   \<comment> \<open>The order of the premises presupposes that @{term C} is rigid;
```
```  1088     @{term A} may be flexible.\<close>
```
```  1089   by auto
```
```  1090
```
```  1091 lemma UnionE [elim!]:
```
```  1092   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
```
```  1093   by auto
```
```  1094
```
```  1095 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
```
```  1096   by (fact Sup_upper)
```
```  1097
```
```  1098 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
```
```  1099   by (fact Sup_least)
```
```  1100
```
```  1101 lemma Union_empty: "\<Union>{} = {}"
```
```  1102   by (fact Sup_empty) (* already simp *)
```
```  1103
```
```  1104 lemma Union_UNIV: "\<Union>UNIV = UNIV"
```
```  1105   by (fact Sup_UNIV) (* already simp *)
```
```  1106
```
```  1107 lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
```
```  1108   by (fact Sup_insert) (* already simp *)
```
```  1109
```
```  1110 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
```
```  1111   by (fact Sup_union_distrib)
```
```  1112
```
```  1113 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
```
```  1114   by (fact Sup_inter_less_eq)
```
```  1115
```
```  1116 lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1117   by (fact Sup_bot_conv) (* already simp *)
```
```  1118
```
```  1119 lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1120   by (fact Sup_bot_conv) (* already simp *)
```
```  1121
```
```  1122 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
```
```  1123   by blast
```
```  1124
```
```  1125 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
```
```  1126   by blast
```
```  1127
```
```  1128 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
```
```  1129   by (fact Sup_subset_mono)
```
```  1130
```
```  1131
```
```  1132 subsubsection \<open>Unions of families\<close>
```
```  1133
```
```  1134 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
```
```  1135   where "UNION \<equiv> SUPREMUM"
```
```  1136
```
```  1137 text \<open>
```
```  1138   Note: must use name @{const UNION} here instead of \<open>UN\<close>
```
```  1139   to allow the following syntax coexist with the plain constant name.
```
```  1140 \<close>
```
```  1141
```
```  1142 syntax (ASCII)
```
```  1143   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
```
```  1144   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
```
```  1145
```
```  1146 syntax (latex output)
```
```  1147   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```  1148   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```  1149
```
```  1150 syntax
```
```  1151   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
```
```  1152   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1153
```
```  1154 translations
```
```  1155   "\<Union>x y. B"   \<rightleftharpoons> "\<Union>x. \<Union>y. B"
```
```  1156   "\<Union>x. B"     \<rightleftharpoons> "CONST UNION CONST UNIV (\<lambda>x. B)"
```
```  1157   "\<Union>x. B"     \<rightleftharpoons> "\<Union>x \<in> CONST UNIV. B"
```
```  1158   "\<Union>x\<in>A. B"   \<rightleftharpoons> "CONST UNION A (\<lambda>x. B)"
```
```  1159
```
```  1160 text \<open>
```
```  1161   Note the difference between ordinary syntax of indexed
```
```  1162   unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>)
```
```  1163   and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}.
```
```  1164 \<close>
```
```  1165
```
```  1166 print_translation \<open>
```
```  1167   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
```
```  1168 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
```
```  1169
```
```  1170 lemma UNION_eq:
```
```  1171   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
```
```  1172   by (auto intro!: SUP_eqI)
```
```  1173
```
```  1174 lemma bind_UNION [code]:
```
```  1175   "Set.bind A f = UNION A f"
```
```  1176   by (simp add: bind_def UNION_eq)
```
```  1177
```
```  1178 lemma member_bind [simp]:
```
```  1179   "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
```
```  1180   by (simp add: bind_UNION)
```
```  1181
```
```  1182 lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
```
```  1183   by blast
```
```  1184
```
```  1185 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
```
```  1186   using Union_iff [of _ "B ` A"] by simp
```
```  1187
```
```  1188 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
```
```  1189   \<comment> \<open>The order of the premises presupposes that @{term A} is rigid;
```
```  1190     @{term b} may be flexible.\<close>
```
```  1191   by auto
```
```  1192
```
```  1193 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"
```
```  1194   by auto
```
```  1195
```
```  1196 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
```
```  1197   by (fact SUP_upper)
```
```  1198
```
```  1199 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
```
```  1200   by (fact SUP_least)
```
```  1201
```
```  1202 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
```
```  1203   by blast
```
```  1204
```
```  1205 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
```
```  1206   by blast
```
```  1207
```
```  1208 lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
```
```  1209   by (fact SUP_empty)
```
```  1210
```
```  1211 lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
```
```  1212   by (fact SUP_bot) (* already simp *)
```
```  1213
```
```  1214 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
```
```  1215   by (fact SUP_absorb)
```
```  1216
```
```  1217 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
```
```  1218   by (fact SUP_insert)
```
```  1219
```
```  1220 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)"
```
```  1221   by (fact SUP_union)
```
```  1222
```
```  1223 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)"
```
```  1224   by blast
```
```  1225
```
```  1226 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
```
```  1227   by (fact SUP_le_iff)
```
```  1228
```
```  1229 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
```
```  1230   by (fact SUP_constant)
```
```  1231
```
```  1232 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
```
```  1233   by blast
```
```  1234
```
```  1235 lemma UNION_empty_conv:
```
```  1236   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1237   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1238   by (fact SUP_bot_conv)+ (* already simp *)
```
```  1239
```
```  1240 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
```
```  1241   by blast
```
```  1242
```
```  1243 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
```
```  1244   by blast
```
```  1245
```
```  1246 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
```
```  1247   by blast
```
```  1248
```
```  1249 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
```
```  1250   by safe (auto simp add: if_split_mem2)
```
```  1251
```
```  1252 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
```
```  1253   by (fact SUP_UNIV_bool_expand)
```
```  1254
```
```  1255 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
```
```  1256   by blast
```
```  1257
```
```  1258 lemma UN_mono:
```
```  1259   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
```
```  1260     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
```
```  1261   by (fact SUP_subset_mono)
```
```  1262
```
```  1263 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
```
```  1264   by blast
```
```  1265
```
```  1266 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
```
```  1267   by blast
```
```  1268
```
```  1269 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
```
```  1270   \<comment> \<open>NOT suitable for rewriting\<close>
```
```  1271   by blast
```
```  1272
```
```  1273 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
```
```  1274   by blast
```
```  1275
```
```  1276 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
```
```  1277   by blast
```
```  1278
```
```  1279 lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on (op ` f) A"
```
```  1280   unfolding inj_on_def by blast
```
```  1281
```
```  1282 subsubsection \<open>Distributive laws\<close>
```
```  1283
```
```  1284 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
```
```  1285   by (fact inf_Sup)
```
```  1286
```
```  1287 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
```
```  1288   by (fact sup_Inf)
```
```  1289
```
```  1290 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
```
```  1291   by (fact Sup_inf)
```
```  1292
```
```  1293 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)"
```
```  1294   by (rule sym) (rule INF_inf_distrib)
```
```  1295
```
```  1296 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)"
```
```  1297   by (rule sym) (rule SUP_sup_distrib)
```
```  1298
```
```  1299 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" \<comment> \<open>FIXME drop\<close>
```
```  1300   by (simp add: INT_Int_distrib)
```
```  1301
```
```  1302 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" \<comment> \<open>FIXME drop\<close>
```
```  1303   \<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close>
```
```  1304   \<comment> \<open>Union of a family of unions\<close>
```
```  1305   by (simp add: UN_Un_distrib)
```
```  1306
```
```  1307 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
```
```  1308   by (fact sup_INF)
```
```  1309
```
```  1310 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
```
```  1311   \<comment> \<open>Halmos, Naive Set Theory, page 35.\<close>
```
```  1312   by (fact inf_SUP)
```
```  1313
```
```  1314 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)"
```
```  1315   by (fact SUP_inf_distrib2)
```
```  1316
```
```  1317 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)"
```
```  1318   by (fact INF_sup_distrib2)
```
```  1319
```
```  1320 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
```
```  1321   by (fact Sup_inf_eq_bot_iff)
```
```  1322
```
```  1323 lemma SUP_UNION: "(SUP x:(UN y:A. g y). f x) = (SUP y:A. SUP x:g y. f x :: _ :: complete_lattice)"
```
```  1324 by(rule order_antisym)(blast intro: SUP_least SUP_upper2)+
```
```  1325
```
```  1326 subsection \<open>Injections and bijections\<close>
```
```  1327
```
```  1328 lemma inj_on_Inter:
```
```  1329   "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
```
```  1330   unfolding inj_on_def by blast
```
```  1331
```
```  1332 lemma inj_on_INTER:
```
```  1333   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)"
```
```  1334   unfolding inj_on_def by safe simp
```
```  1335
```
```  1336 lemma inj_on_UNION_chain:
```
```  1337   assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
```
```  1338          INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1339   shows "inj_on f (\<Union>i \<in> I. A i)"
```
```  1340 proof -
```
```  1341   {
```
```  1342     fix i j x y
```
```  1343     assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
```
```  1344       and ***: "f x = f y"
```
```  1345     have "x = y"
```
```  1346     proof -
```
```  1347       {
```
```  1348         assume "A i \<le> A j"
```
```  1349         with ** have "x \<in> A j" by auto
```
```  1350         with INJ * ** *** have ?thesis
```
```  1351         by(auto simp add: inj_on_def)
```
```  1352       }
```
```  1353       moreover
```
```  1354       {
```
```  1355         assume "A j \<le> A i"
```
```  1356         with ** have "y \<in> A i" by auto
```
```  1357         with INJ * ** *** have ?thesis
```
```  1358         by(auto simp add: inj_on_def)
```
```  1359       }
```
```  1360       ultimately show ?thesis using CH * by blast
```
```  1361     qed
```
```  1362   }
```
```  1363   then show ?thesis by (unfold inj_on_def UNION_eq) auto
```
```  1364 qed
```
```  1365
```
```  1366 lemma bij_betw_UNION_chain:
```
```  1367   assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
```
```  1368          BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
```
```  1369   shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)"
```
```  1370 proof (unfold bij_betw_def, auto)
```
```  1371   have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1372   using BIJ bij_betw_def[of f] by auto
```
```  1373   thus "inj_on f (\<Union>i \<in> I. A i)"
```
```  1374   using CH inj_on_UNION_chain[of I A f] by auto
```
```  1375 next
```
```  1376   fix i x
```
```  1377   assume *: "i \<in> I" "x \<in> A i"
```
```  1378   hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
```
```  1379   thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
```
```  1380 next
```
```  1381   fix i x'
```
```  1382   assume *: "i \<in> I" "x' \<in> A' i"
```
```  1383   hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
```
```  1384   then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
```
```  1385     using * by blast
```
```  1386   then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
```
```  1387 qed
```
```  1388
```
```  1389 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```  1390 lemma image_INT:
```
```  1391    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```  1392     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1393   by (simp add: inj_on_def, auto) blast
```
```  1394
```
```  1395 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1396   apply (simp add: bij_def)
```
```  1397   apply (simp add: inj_on_def surj_def)
```
```  1398   apply auto
```
```  1399   apply blast
```
```  1400   done
```
```  1401
```
```  1402 lemma UNION_fun_upd:
```
```  1403   "UNION J (A(i := B)) = UNION (J - {i}) A \<union> (if i \<in> J then B else {})"
```
```  1404   by (auto simp add: set_eq_iff)
```
```  1405
```
```  1406
```
```  1407 subsubsection \<open>Complement\<close>
```
```  1408
```
```  1409 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
```
```  1410   by (fact uminus_INF)
```
```  1411
```
```  1412 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
```
```  1413   by (fact uminus_SUP)
```
```  1414
```
```  1415
```
```  1416 subsubsection \<open>Miniscoping and maxiscoping\<close>
```
```  1417
```
```  1418 text \<open>\medskip Miniscoping: pushing in quantifiers and big Unions
```
```  1419            and Intersections.\<close>
```
```  1420
```
```  1421 lemma UN_simps [simp]:
```
```  1422   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
```
```  1423   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
```
```  1424   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
```
```  1425   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
```
```  1426   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
```
```  1427   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
```
```  1428   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
```
```  1429   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
```
```  1430   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
```
```  1431   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
```
```  1432   by auto
```
```  1433
```
```  1434 lemma INT_simps [simp]:
```
```  1435   "\<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)"
```
```  1436   "\<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))"
```
```  1437   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
```
```  1438   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
```
```  1439   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
```
```  1440   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
```
```  1441   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
```
```  1442   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
```
```  1443   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
```
```  1444   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
```
```  1445   by auto
```
```  1446
```
```  1447 lemma UN_ball_bex_simps [simp]:
```
```  1448   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
```
```  1449   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
```
```  1450   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
```
```  1451   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
```
```  1452   by auto
```
```  1453
```
```  1454
```
```  1455 text \<open>\medskip Maxiscoping: pulling out big Unions and Intersections.\<close>
```
```  1456
```
```  1457 lemma UN_extend_simps:
```
```  1458   "\<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)))"
```
```  1459   "\<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))"
```
```  1460   "\<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))"
```
```  1461   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
```
```  1462   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
```
```  1463   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
```
```  1464   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
```
```  1465   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
```
```  1466   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
```
```  1467   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
```
```  1468   by auto
```
```  1469
```
```  1470 lemma INT_extend_simps:
```
```  1471   "\<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))"
```
```  1472   "\<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))"
```
```  1473   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
```
```  1474   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
```
```  1475   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
```
```  1476   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
```
```  1477   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
```
```  1478   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
```
```  1479   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
```
```  1480   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
```
```  1481   by auto
```
```  1482
```
```  1483 text \<open>Finally\<close>
```
```  1484
```
```  1485 no_notation
```
```  1486   less_eq (infix "\<sqsubseteq>" 50) and
```
```  1487   less (infix "\<sqsubset>" 50)
```
```  1488
```
```  1489 lemmas mem_simps =
```
```  1490   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
```
```  1491   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
```
```  1492   \<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close>
```
```  1493
```
```  1494 end
```
```  1495
```