src/HOL/Complete_Lattices.thy
 author haftmann Fri Oct 10 19:55:32 2014 +0200 (2014-10-10) changeset 58646 cd63a4b12a33 parent 57448 159e45728ceb child 58889 5b7a9633cfa8 permissions -rw-r--r--
specialized specification: avoid trivial instances
```     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 Fun
```
```     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 INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    21   INF_def: "INFIMUM A f = \<Sqinter>(f ` A)"
```
```    22
```
```    23 lemma Inf_image_eq [simp]:
```
```    24   "\<Sqinter>(f ` A) = INFIMUM A f"
```
```    25   by (simp add: INF_def)
```
```    26
```
```    27 lemma INF_image [simp]:
```
```    28   "INFIMUM (f ` A) g = INFIMUM A (g \<circ> f)"
```
```    29   by (simp only: INF_def image_comp)
```
```    30
```
```    31 lemma INF_identity_eq [simp]:
```
```    32   "INFIMUM A (\<lambda>x. x) = \<Sqinter>A"
```
```    33   by (simp add: INF_def)
```
```    34
```
```    35 lemma INF_id_eq [simp]:
```
```    36   "INFIMUM A id = \<Sqinter>A"
```
```    37   by (simp add: id_def)
```
```    38
```
```    39 lemma INF_cong:
```
```    40   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
```
```    41   by (simp add: INF_def image_def)
```
```    42
```
```    43 lemma strong_INF_cong [cong]:
```
```    44   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> INFIMUM A C = INFIMUM B D"
```
```    45   unfolding simp_implies_def by (fact INF_cong)
```
```    46
```
```    47 end
```
```    48
```
```    49 class Sup =
```
```    50   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```    51 begin
```
```    52
```
```    53 definition SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    54   SUP_def: "SUPREMUM A f = \<Squnion>(f ` A)"
```
```    55
```
```    56 lemma Sup_image_eq [simp]:
```
```    57   "\<Squnion>(f ` A) = SUPREMUM A f"
```
```    58   by (simp add: SUP_def)
```
```    59
```
```    60 lemma SUP_image [simp]:
```
```    61   "SUPREMUM (f ` A) g = SUPREMUM A (g \<circ> f)"
```
```    62   by (simp only: SUP_def image_comp)
```
```    63
```
```    64 lemma SUP_identity_eq [simp]:
```
```    65   "SUPREMUM A (\<lambda>x. x) = \<Squnion>A"
```
```    66   by (simp add: SUP_def)
```
```    67
```
```    68 lemma SUP_id_eq [simp]:
```
```    69   "SUPREMUM A id = \<Squnion>A"
```
```    70   by (simp add: id_def)
```
```    71
```
```    72 lemma SUP_cong:
```
```    73   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
```
```    74   by (simp add: SUP_def image_def)
```
```    75
```
```    76 lemma strong_SUP_cong [cong]:
```
```    77   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> SUPREMUM A C = SUPREMUM B D"
```
```    78   unfolding simp_implies_def by (fact SUP_cong)
```
```    79
```
```    80 end
```
```    81
```
```    82 text {*
```
```    83   Note: must use names @{const INFIMUM} and @{const SUPREMUM} here instead of
```
```    84   @{text INF} and @{text SUP} to allow the following syntax coexist
```
```    85   with the plain constant names.
```
```    86 *}
```
```    87
```
```    88 syntax
```
```    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 (xsymbols)
```
```    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   "INF x y. B"   == "INF x. INF y. B"
```
```   102   "INF x. B"     == "CONST INFIMUM CONST UNIV (%x. B)"
```
```   103   "INF x. B"     == "INF x:CONST UNIV. B"
```
```   104   "INF x:A. B"   == "CONST INFIMUM A (%x. B)"
```
```   105   "SUP x y. B"   == "SUP x. SUP y. B"
```
```   106   "SUP x. B"     == "CONST SUPREMUM CONST UNIV (%x. B)"
```
```   107   "SUP x. B"     == "SUP x:CONST UNIV. B"
```
```   108   "SUP x:A. B"   == "CONST SUPREMUM A (%x. B)"
```
```   109
```
```   110 print_translation {*
```
```   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 *} -- {* to avoid eta-contraction of body *}
```
```   114
```
```   115 subsection {* Abstract complete lattices *}
```
```   116
```
```   117 text {* A complete lattice always has a bottom and a top,
```
```   118 so we include them into the following type class,
```
```   119 along with assumptions that define bottom and top
```
```   120 in terms of infimum and supremum. *}
```
```   121
```
```   122 class complete_lattice = lattice + Inf + Sup + bot + top +
```
```   123   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```   124      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```   125   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```   126      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```   127   assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
```
```   128   assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
```
```   129 begin
```
```   130
```
```   131 subclass bounded_lattice
```
```   132 proof
```
```   133   fix a
```
```   134   show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
```
```   135   show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
```
```   136 qed
```
```   137
```
```   138 lemma dual_complete_lattice:
```
```   139   "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   140   by (auto intro!: class.complete_lattice.intro dual_lattice)
```
```   141     (unfold_locales, (fact Inf_empty Sup_empty
```
```   142         Sup_upper Sup_least Inf_lower Inf_greatest)+)
```
```   143
```
```   144 end
```
```   145
```
```   146 context complete_lattice
```
```   147 begin
```
```   148
```
```   149 lemma INF_foundation_dual:
```
```   150   "Sup.SUPREMUM Inf = INFIMUM"
```
```   151   by (simp add: fun_eq_iff Sup.SUP_def)
```
```   152
```
```   153 lemma SUP_foundation_dual:
```
```   154   "Inf.INFIMUM Sup = SUPREMUM"
```
```   155   by (simp add: fun_eq_iff Inf.INF_def)
```
```   156
```
```   157 lemma Sup_eqI:
```
```   158   "(\<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"
```
```   159   by (blast intro: antisym Sup_least Sup_upper)
```
```   160
```
```   161 lemma Inf_eqI:
```
```   162   "(\<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"
```
```   163   by (blast intro: antisym Inf_greatest Inf_lower)
```
```   164
```
```   165 lemma SUP_eqI:
```
```   166   "(\<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"
```
```   167   using Sup_eqI [of "f ` A" x] by auto
```
```   168
```
```   169 lemma INF_eqI:
```
```   170   "(\<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"
```
```   171   using Inf_eqI [of "f ` A" x] by auto
```
```   172
```
```   173 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
```
```   174   using Inf_lower [of _ "f ` A"] by simp
```
```   175
```
```   176 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
```
```   177   using Inf_greatest [of "f ` A"] by auto
```
```   178
```
```   179 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   180   using Sup_upper [of _ "f ` A"] by simp
```
```   181
```
```   182 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
```
```   183   using Sup_least [of "f ` A"] by auto
```
```   184
```
```   185 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
```
```   186   using Inf_lower [of u A] by auto
```
```   187
```
```   188 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
```
```   189   using INF_lower [of i A f] by auto
```
```   190
```
```   191 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
```
```   192   using Sup_upper [of u A] by auto
```
```   193
```
```   194 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   195   using SUP_upper [of i A f] by auto
```
```   196
```
```   197 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
```
```   198   by (auto intro: Inf_greatest dest: Inf_lower)
```
```   199
```
```   200 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
```
```   201   using le_Inf_iff [of _ "f ` A"] by simp
```
```   202
```
```   203 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
```
```   204   by (auto intro: Sup_least dest: Sup_upper)
```
```   205
```
```   206 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
```
```   207   using Sup_le_iff [of "f ` A"] by simp
```
```   208
```
```   209 lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   210   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
```
```   211
```
```   212 lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFIMUM A f"
```
```   213   unfolding INF_def Inf_insert by simp
```
```   214
```
```   215 lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   216   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
```
```   217
```
```   218 lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPREMUM A f"
```
```   219   unfolding SUP_def Sup_insert by simp
```
```   220
```
```   221 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
```
```   222   by (simp add: INF_def)
```
```   223
```
```   224 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
```
```   225   by (simp add: SUP_def)
```
```   226
```
```   227 lemma Inf_UNIV [simp]:
```
```   228   "\<Sqinter>UNIV = \<bottom>"
```
```   229   by (auto intro!: antisym Inf_lower)
```
```   230
```
```   231 lemma Sup_UNIV [simp]:
```
```   232   "\<Squnion>UNIV = \<top>"
```
```   233   by (auto intro!: antisym Sup_upper)
```
```   234
```
```   235 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
```
```   236   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   237
```
```   238 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
```
```   239   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   240
```
```   241 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   242   by (auto intro: Inf_greatest Inf_lower)
```
```   243
```
```   244 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   245   by (auto intro: Sup_least Sup_upper)
```
```   246
```
```   247 lemma Inf_mono:
```
```   248   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
```
```   249   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   250 proof (rule Inf_greatest)
```
```   251   fix b assume "b \<in> B"
```
```   252   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
```
```   253   from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
```
```   254   with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
```
```   255 qed
```
```   256
```
```   257 lemma INF_mono:
```
```   258   "(\<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)"
```
```   259   using Inf_mono [of "g ` B" "f ` A"] by auto
```
```   260
```
```   261 lemma Sup_mono:
```
```   262   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
```
```   263   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   264 proof (rule Sup_least)
```
```   265   fix a assume "a \<in> A"
```
```   266   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
```
```   267   from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
```
```   268   with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
```
```   269 qed
```
```   270
```
```   271 lemma SUP_mono:
```
```   272   "(\<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)"
```
```   273   using Sup_mono [of "f ` A" "g ` B"] by auto
```
```   274
```
```   275 lemma INF_superset_mono:
```
```   276   "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)"
```
```   277   -- {* The last inclusion is POSITIVE! *}
```
```   278   by (blast intro: INF_mono dest: subsetD)
```
```   279
```
```   280 lemma SUP_subset_mono:
```
```   281   "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)"
```
```   282   by (blast intro: SUP_mono dest: subsetD)
```
```   283
```
```   284 lemma Inf_less_eq:
```
```   285   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
```
```   286     and "A \<noteq> {}"
```
```   287   shows "\<Sqinter>A \<sqsubseteq> u"
```
```   288 proof -
```
```   289   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   290   moreover from `v \<in> A` assms(1) have "v \<sqsubseteq> u" by blast
```
```   291   ultimately show ?thesis by (rule Inf_lower2)
```
```   292 qed
```
```   293
```
```   294 lemma less_eq_Sup:
```
```   295   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
```
```   296     and "A \<noteq> {}"
```
```   297   shows "u \<sqsubseteq> \<Squnion>A"
```
```   298 proof -
```
```   299   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   300   moreover from `v \<in> A` assms(1) have "u \<sqsubseteq> v" by blast
```
```   301   ultimately show ?thesis by (rule Sup_upper2)
```
```   302 qed
```
```   303
```
```   304 lemma SUP_eq:
```
```   305   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
```
```   306   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
```
```   307   shows "(\<Squnion>i\<in>A. f i) = (\<Squnion>j\<in>B. g j)"
```
```   308   by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
```
```   309
```
```   310 lemma INF_eq:
```
```   311   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
```
```   312   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
```
```   313   shows "(\<Sqinter>i\<in>A. f i) = (\<Sqinter>j\<in>B. g j)"
```
```   314   by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
```
```   315
```
```   316 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
```
```   317   by (auto intro: Inf_greatest Inf_lower)
```
```   318
```
```   319 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
```
```   320   by (auto intro: Sup_least Sup_upper)
```
```   321
```
```   322 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
```
```   323   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
```
```   324
```
```   325 lemma INF_union:
```
```   326   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
```
```   327   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
```
```   328
```
```   329 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
```
```   330   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
```
```   331
```
```   332 lemma SUP_union:
```
```   333   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
```
```   334   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
```
```   335
```
```   336 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)"
```
```   337   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
```
```   338
```
```   339 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")
```
```   340 proof (rule antisym)
```
```   341   show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
```
```   342 next
```
```   343   show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
```
```   344 qed
```
```   345
```
```   346 lemma Inf_top_conv [simp]:
```
```   347   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   348   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   349 proof -
```
```   350   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   351   proof
```
```   352     assume "\<forall>x\<in>A. x = \<top>"
```
```   353     then have "A = {} \<or> A = {\<top>}" by auto
```
```   354     then show "\<Sqinter>A = \<top>" by auto
```
```   355   next
```
```   356     assume "\<Sqinter>A = \<top>"
```
```   357     show "\<forall>x\<in>A. x = \<top>"
```
```   358     proof (rule ccontr)
```
```   359       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
```
```   360       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
```
```   361       then obtain B where "A = insert x B" by blast
```
```   362       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
```
```   363     qed
```
```   364   qed
```
```   365   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
```
```   366 qed
```
```   367
```
```   368 lemma INF_top_conv [simp]:
```
```   369   "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   370   "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   371   using Inf_top_conv [of "B ` A"] by simp_all
```
```   372
```
```   373 lemma Sup_bot_conv [simp]:
```
```   374   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
```
```   375   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
```
```   376   using dual_complete_lattice
```
```   377   by (rule complete_lattice.Inf_top_conv)+
```
```   378
```
```   379 lemma SUP_bot_conv [simp]:
```
```   380  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   381  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   382   using Sup_bot_conv [of "B ` A"] by simp_all
```
```   383
```
```   384 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
```
```   385   by (auto intro: antisym INF_lower INF_greatest)
```
```   386
```
```   387 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
```
```   388   by (auto intro: antisym SUP_upper SUP_least)
```
```   389
```
```   390 lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
```
```   391   by (cases "A = {}") simp_all
```
```   392
```
```   393 lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
```
```   394   by (cases "A = {}") simp_all
```
```   395
```
```   396 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)"
```
```   397   by (iprover intro: INF_lower INF_greatest order_trans antisym)
```
```   398
```
```   399 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)"
```
```   400   by (iprover intro: SUP_upper SUP_least order_trans antisym)
```
```   401
```
```   402 lemma INF_absorb:
```
```   403   assumes "k \<in> I"
```
```   404   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
```
```   405 proof -
```
```   406   from assms obtain J where "I = insert k J" by blast
```
```   407   then show ?thesis by simp
```
```   408 qed
```
```   409
```
```   410 lemma SUP_absorb:
```
```   411   assumes "k \<in> I"
```
```   412   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
```
```   413 proof -
```
```   414   from assms obtain J where "I = insert k J" by blast
```
```   415   then show ?thesis by simp
```
```   416 qed
```
```   417
```
```   418 lemma INF_inf_const1:
```
```   419   "I \<noteq> {} \<Longrightarrow> (INF i:I. inf x (f i)) = inf x (INF i:I. f i)"
```
```   420   by (intro antisym INF_greatest inf_mono order_refl INF_lower)
```
```   421      (auto intro: INF_lower2 le_infI2 intro!: INF_mono)
```
```   422
```
```   423 lemma INF_inf_const2:
```
```   424   "I \<noteq> {} \<Longrightarrow> (INF i:I. inf (f i) x) = inf (INF i:I. f i) x"
```
```   425   using INF_inf_const1[of I x f] by (simp add: inf_commute)
```
```   426
```
```   427 lemma INF_constant:
```
```   428   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
```
```   429   by simp
```
```   430
```
```   431 lemma SUP_constant:
```
```   432   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
```
```   433   by simp
```
```   434
```
```   435 lemma less_INF_D:
```
```   436   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
```
```   437 proof -
```
```   438   note `y < (\<Sqinter>i\<in>A. f i)`
```
```   439   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
```
```   440     by (rule INF_lower)
```
```   441   finally show "y < f i" .
```
```   442 qed
```
```   443
```
```   444 lemma SUP_lessD:
```
```   445   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
```
```   446 proof -
```
```   447   have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
```
```   448     by (rule SUP_upper)
```
```   449   also note `(\<Squnion>i\<in>A. f i) < y`
```
```   450   finally show "f i < y" .
```
```   451 qed
```
```   452
```
```   453 lemma INF_UNIV_bool_expand:
```
```   454   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
```
```   455   by (simp add: UNIV_bool inf_commute)
```
```   456
```
```   457 lemma SUP_UNIV_bool_expand:
```
```   458   "(\<Squnion>b. A b) = A True \<squnion> A False"
```
```   459   by (simp add: UNIV_bool sup_commute)
```
```   460
```
```   461 lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
```
```   462   by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
```
```   463
```
```   464 lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFIMUM A f \<le> SUPREMUM A f"
```
```   465   using Inf_le_Sup [of "f ` A"] by simp
```
```   466
```
```   467 lemma INF_eq_const:
```
```   468   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFIMUM I f = x"
```
```   469   by (auto intro: INF_eqI)
```
```   470
```
```   471 lemma SUP_eq_const:
```
```   472   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPREMUM I f = x"
```
```   473   by (auto intro: SUP_eqI)
```
```   474
```
```   475 lemma INF_eq_iff:
```
```   476   "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)"
```
```   477   using INF_eq_const [of I f c] INF_lower [of _ I f]
```
```   478   by (auto intro: antisym cong del: strong_INF_cong)
```
```   479
```
```   480 lemma SUP_eq_iff:
```
```   481   "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)"
```
```   482   using SUP_eq_const [of I f c] SUP_upper [of _ I f]
```
```   483   by (auto intro: antisym cong del: strong_SUP_cong)
```
```   484
```
```   485 end
```
```   486
```
```   487 class complete_distrib_lattice = complete_lattice +
```
```   488   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
```
```   489   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   490 begin
```
```   491
```
```   492 lemma sup_INF:
```
```   493   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
```
```   494   by (simp only: INF_def sup_Inf image_image)
```
```   495
```
```   496 lemma inf_SUP:
```
```   497   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
```
```   498   by (simp only: SUP_def inf_Sup image_image)
```
```   499
```
```   500 lemma dual_complete_distrib_lattice:
```
```   501   "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   502   apply (rule class.complete_distrib_lattice.intro)
```
```   503   apply (fact dual_complete_lattice)
```
```   504   apply (rule class.complete_distrib_lattice_axioms.intro)
```
```   505   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
```
```   506   done
```
```   507
```
```   508 subclass distrib_lattice proof
```
```   509   fix a b c
```
```   510   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
```
```   511   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
```
```   512 qed
```
```   513
```
```   514 lemma Inf_sup:
```
```   515   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
```
```   516   by (simp add: sup_Inf sup_commute)
```
```   517
```
```   518 lemma Sup_inf:
```
```   519   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
```
```   520   by (simp add: inf_Sup inf_commute)
```
```   521
```
```   522 lemma INF_sup:
```
```   523   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
```
```   524   by (simp add: sup_INF sup_commute)
```
```   525
```
```   526 lemma SUP_inf:
```
```   527   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
```
```   528   by (simp add: inf_SUP inf_commute)
```
```   529
```
```   530 lemma Inf_sup_eq_top_iff:
```
```   531   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
```
```   532   by (simp only: Inf_sup INF_top_conv)
```
```   533
```
```   534 lemma Sup_inf_eq_bot_iff:
```
```   535   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
```
```   536   by (simp only: Sup_inf SUP_bot_conv)
```
```   537
```
```   538 lemma INF_sup_distrib2:
```
```   539   "(\<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)"
```
```   540   by (subst INF_commute) (simp add: sup_INF INF_sup)
```
```   541
```
```   542 lemma SUP_inf_distrib2:
```
```   543   "(\<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)"
```
```   544   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
```
```   545
```
```   546 context
```
```   547   fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
```
```   548   assumes "mono f"
```
```   549 begin
```
```   550
```
```   551 lemma mono_Inf:
```
```   552   shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
```
```   553   using `mono f` by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
```
```   554
```
```   555 lemma mono_Sup:
```
```   556   shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
```
```   557   using `mono f` by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
```
```   558
```
```   559 end
```
```   560
```
```   561 end
```
```   562
```
```   563 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
```
```   564 begin
```
```   565
```
```   566 lemma dual_complete_boolean_algebra:
```
```   567   "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
```
```   568   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
```
```   569
```
```   570 lemma uminus_Inf:
```
```   571   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
```
```   572 proof (rule antisym)
```
```   573   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
```
```   574     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
```
```   575   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
```
```   576     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
```
```   577 qed
```
```   578
```
```   579 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
```
```   580   by (simp only: INF_def SUP_def uminus_Inf image_image)
```
```   581
```
```   582 lemma uminus_Sup:
```
```   583   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
```
```   584 proof -
```
```   585   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
```
```   586   then show ?thesis by simp
```
```   587 qed
```
```   588
```
```   589 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
```
```   590   by (simp only: INF_def SUP_def uminus_Sup image_image)
```
```   591
```
```   592 end
```
```   593
```
```   594 class complete_linorder = linorder + complete_lattice
```
```   595 begin
```
```   596
```
```   597 lemma dual_complete_linorder:
```
```   598   "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   599   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
```
```   600
```
```   601 lemma complete_linorder_inf_min: "inf = min"
```
```   602   by (auto intro: antisym simp add: min_def fun_eq_iff)
```
```   603
```
```   604 lemma complete_linorder_sup_max: "sup = max"
```
```   605   by (auto intro: antisym simp add: max_def fun_eq_iff)
```
```   606
```
```   607 lemma Inf_less_iff:
```
```   608   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
```
```   609   unfolding not_le [symmetric] le_Inf_iff by auto
```
```   610
```
```   611 lemma INF_less_iff:
```
```   612   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
```
```   613   using Inf_less_iff [of "f ` A"] by simp
```
```   614
```
```   615 lemma less_Sup_iff:
```
```   616   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
```
```   617   unfolding not_le [symmetric] Sup_le_iff by auto
```
```   618
```
```   619 lemma less_SUP_iff:
```
```   620   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
```
```   621   using less_Sup_iff [of _ "f ` A"] by simp
```
```   622
```
```   623 lemma Sup_eq_top_iff [simp]:
```
```   624   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
```
```   625 proof
```
```   626   assume *: "\<Squnion>A = \<top>"
```
```   627   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
```
```   628   proof (intro allI impI)
```
```   629     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
```
```   630       unfolding less_Sup_iff by auto
```
```   631   qed
```
```   632 next
```
```   633   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
```
```   634   show "\<Squnion>A = \<top>"
```
```   635   proof (rule ccontr)
```
```   636     assume "\<Squnion>A \<noteq> \<top>"
```
```   637     with top_greatest [of "\<Squnion>A"]
```
```   638     have "\<Squnion>A < \<top>" unfolding le_less by auto
```
```   639     then have "\<Squnion>A < \<Squnion>A"
```
```   640       using * unfolding less_Sup_iff by auto
```
```   641     then show False by auto
```
```   642   qed
```
```   643 qed
```
```   644
```
```   645 lemma SUP_eq_top_iff [simp]:
```
```   646   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
```
```   647   using Sup_eq_top_iff [of "f ` A"] by simp
```
```   648
```
```   649 lemma Inf_eq_bot_iff [simp]:
```
```   650   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
```
```   651   using dual_complete_linorder
```
```   652   by (rule complete_linorder.Sup_eq_top_iff)
```
```   653
```
```   654 lemma INF_eq_bot_iff [simp]:
```
```   655   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
```
```   656   using Inf_eq_bot_iff [of "f ` A"] by simp
```
```   657
```
```   658 lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
```
```   659 proof safe
```
```   660   fix y assume "x \<ge> \<Sqinter>A" "y > x"
```
```   661   then have "y > \<Sqinter>A" by auto
```
```   662   then show "\<exists>a\<in>A. y > a"
```
```   663     unfolding Inf_less_iff .
```
```   664 qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
```
```   665
```
```   666 lemma INF_le_iff:
```
```   667   "INFIMUM A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
```
```   668   using Inf_le_iff [of "f ` A"] by simp
```
```   669
```
```   670 lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
```
```   671 proof safe
```
```   672   fix y assume "x \<le> \<Squnion>A" "y < x"
```
```   673   then have "y < \<Squnion>A" by auto
```
```   674   then show "\<exists>a\<in>A. y < a"
```
```   675     unfolding less_Sup_iff .
```
```   676 qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
```
```   677
```
```   678 lemma le_SUP_iff: "x \<le> SUPREMUM A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
```
```   679   using le_Sup_iff [of _ "f ` A"] by simp
```
```   680
```
```   681 subclass complete_distrib_lattice
```
```   682 proof
```
```   683   fix a and B
```
```   684   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   685     by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
```
```   686       (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
```
```   687         le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
```
```   688 qed
```
```   689
```
```   690 end
```
```   691
```
```   692
```
```   693 subsection {* Complete lattice on @{typ bool} *}
```
```   694
```
```   695 instantiation bool :: complete_lattice
```
```   696 begin
```
```   697
```
```   698 definition
```
```   699   [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
```
```   700
```
```   701 definition
```
```   702   [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
```
```   703
```
```   704 instance proof
```
```   705 qed (auto intro: bool_induct)
```
```   706
```
```   707 end
```
```   708
```
```   709 lemma not_False_in_image_Ball [simp]:
```
```   710   "False \<notin> P ` A \<longleftrightarrow> Ball A P"
```
```   711   by auto
```
```   712
```
```   713 lemma True_in_image_Bex [simp]:
```
```   714   "True \<in> P ` A \<longleftrightarrow> Bex A P"
```
```   715   by auto
```
```   716
```
```   717 lemma INF_bool_eq [simp]:
```
```   718   "INFIMUM = Ball"
```
```   719   by (simp add: fun_eq_iff INF_def)
```
```   720
```
```   721 lemma SUP_bool_eq [simp]:
```
```   722   "SUPREMUM = Bex"
```
```   723   by (simp add: fun_eq_iff SUP_def)
```
```   724
```
```   725 instance bool :: complete_boolean_algebra proof
```
```   726 qed (auto intro: bool_induct)
```
```   727
```
```   728
```
```   729 subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
```
```   730
```
```   731 instantiation "fun" :: (type, Inf) Inf
```
```   732 begin
```
```   733
```
```   734 definition
```
```   735   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
```
```   736
```
```   737 lemma Inf_apply [simp, code]:
```
```   738   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
```
```   739   by (simp add: Inf_fun_def)
```
```   740
```
```   741 instance ..
```
```   742
```
```   743 end
```
```   744
```
```   745 instantiation "fun" :: (type, Sup) Sup
```
```   746 begin
```
```   747
```
```   748 definition
```
```   749   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
```
```   750
```
```   751 lemma Sup_apply [simp, code]:
```
```   752   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
```
```   753   by (simp add: Sup_fun_def)
```
```   754
```
```   755 instance ..
```
```   756
```
```   757 end
```
```   758
```
```   759 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   760 begin
```
```   761
```
```   762 instance proof
```
```   763 qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
```
```   764
```
```   765 end
```
```   766
```
```   767 lemma INF_apply [simp]:
```
```   768   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
```
```   769   using Inf_apply [of "f ` A"] by (simp add: comp_def)
```
```   770
```
```   771 lemma SUP_apply [simp]:
```
```   772   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
```
```   773   using Sup_apply [of "f ` A"] by (simp add: comp_def)
```
```   774
```
```   775 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
```
```   776 qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf fun_eq_iff image_image
```
```   777   simp del: Inf_image_eq Sup_image_eq)
```
```   778
```
```   779 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
```
```   780
```
```   781
```
```   782 subsection {* Complete lattice on unary and binary predicates *}
```
```   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 {* Complete lattice on @{typ "_ set"} *}
```
```   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_def SUP_def Inf_set_def Sup_set_def image_def)
```
```   892
```
```   893
```
```   894 subsubsection {* Inter *}
```
```   895
```
```   896 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
```
```   897   "Inter S \<equiv> \<Sqinter>S"
```
```   898
```
```   899 notation (xsymbols)
```
```   900   Inter  ("\<Inter>_" [900] 900)
```
```   901
```
```   902 lemma Inter_eq:
```
```   903   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
```
```   904 proof (rule set_eqI)
```
```   905   fix x
```
```   906   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
```
```   907     by auto
```
```   908   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
```
```   909     by (simp add: Inf_set_def image_def)
```
```   910 qed
```
```   911
```
```   912 lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
```
```   913   by (unfold Inter_eq) blast
```
```   914
```
```   915 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
```
```   916   by (simp add: Inter_eq)
```
```   917
```
```   918 text {*
```
```   919   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
```
```   920   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
```
```   921   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
```
```   922 *}
```
```   923
```
```   924 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
```
```   925   by auto
```
```   926
```
```   927 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
```
```   928   -- {* ``Classical'' elimination rule -- does not require proving
```
```   929     @{prop "X \<in> C"}. *}
```
```   930   by (unfold Inter_eq) blast
```
```   931
```
```   932 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   933   by (fact Inf_lower)
```
```   934
```
```   935 lemma Inter_subset:
```
```   936   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   937   by (fact Inf_less_eq)
```
```   938
```
```   939 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
```
```   940   by (fact Inf_greatest)
```
```   941
```
```   942 lemma Inter_empty: "\<Inter>{} = UNIV"
```
```   943   by (fact Inf_empty) (* already simp *)
```
```   944
```
```   945 lemma Inter_UNIV: "\<Inter>UNIV = {}"
```
```   946   by (fact Inf_UNIV) (* already simp *)
```
```   947
```
```   948 lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
```
```   949   by (fact Inf_insert) (* already simp *)
```
```   950
```
```   951 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
```
```   952   by (fact less_eq_Inf_inter)
```
```   953
```
```   954 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
```
```   955   by (fact Inf_union_distrib)
```
```   956
```
```   957 lemma Inter_UNIV_conv [simp]:
```
```   958   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   959   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   960   by (fact Inf_top_conv)+
```
```   961
```
```   962 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
```
```   963   by (fact Inf_superset_mono)
```
```   964
```
```   965
```
```   966 subsubsection {* Intersections of families *}
```
```   967
```
```   968 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```   969   "INTER \<equiv> INFIMUM"
```
```   970
```
```   971 text {*
```
```   972   Note: must use name @{const INTER} here instead of @{text INT}
```
```   973   to allow the following syntax coexist with the plain constant name.
```
```   974 *}
```
```   975
```
```   976 syntax
```
```   977   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
```
```   978   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
```
```   979
```
```   980 syntax (xsymbols)
```
```   981   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
```
```   982   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
```
```   983
```
```   984 syntax (latex output)
```
```   985   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   986   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   987
```
```   988 translations
```
```   989   "INT x y. B"  == "INT x. INT y. B"
```
```   990   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
```
```   991   "INT x. B"    == "INT x:CONST UNIV. B"
```
```   992   "INT x:A. B"  == "CONST INTER A (%x. B)"
```
```   993
```
```   994 print_translation {*
```
```   995   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
```
```   996 *} -- {* to avoid eta-contraction of body *}
```
```   997
```
```   998 lemma INTER_eq:
```
```   999   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
```
```  1000   by (auto intro!: INF_eqI)
```
```  1001
```
```  1002 lemma Inter_image_eq:
```
```  1003   "\<Inter>(B ` A) = (\<Inter>x\<in>A. B x)"
```
```  1004   by (fact Inf_image_eq)
```
```  1005
```
```  1006 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
```
```  1007   using Inter_iff [of _ "B ` A"] by simp
```
```  1008
```
```  1009 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
```
```  1010   by (auto simp add: INF_def image_def)
```
```  1011
```
```  1012 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
```
```  1013   by auto
```
```  1014
```
```  1015 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"
```
```  1016   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
```
```  1017   by (auto simp add: INF_def image_def)
```
```  1018
```
```  1019 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
```
```  1020   by blast
```
```  1021
```
```  1022 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
```
```  1023   by blast
```
```  1024
```
```  1025 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
```
```  1026   by (fact INF_lower)
```
```  1027
```
```  1028 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
```
```  1029   by (fact INF_greatest)
```
```  1030
```
```  1031 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
```
```  1032   by (fact INF_empty)
```
```  1033
```
```  1034 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
```
```  1035   by (fact INF_absorb)
```
```  1036
```
```  1037 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
```
```  1038   by (fact le_INF_iff)
```
```  1039
```
```  1040 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
```
```  1041   by (fact INF_insert)
```
```  1042
```
```  1043 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
```
```  1044   by (fact INF_union)
```
```  1045
```
```  1046 lemma INT_insert_distrib:
```
```  1047   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
```
```  1048   by blast
```
```  1049
```
```  1050 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
```
```  1051   by (fact INF_constant)
```
```  1052
```
```  1053 lemma INTER_UNIV_conv:
```
```  1054  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
```
```  1055  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
```
```  1056   by (fact INF_top_conv)+ (* already simp *)
```
```  1057
```
```  1058 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
```
```  1059   by (fact INF_UNIV_bool_expand)
```
```  1060
```
```  1061 lemma INT_anti_mono:
```
```  1062   "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)"
```
```  1063   -- {* The last inclusion is POSITIVE! *}
```
```  1064   by (fact INF_superset_mono)
```
```  1065
```
```  1066 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
```
```  1067   by blast
```
```  1068
```
```  1069 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
```
```  1070   by blast
```
```  1071
```
```  1072
```
```  1073 subsubsection {* Union *}
```
```  1074
```
```  1075 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
```
```  1076   "Union S \<equiv> \<Squnion>S"
```
```  1077
```
```  1078 notation (xsymbols)
```
```  1079   Union  ("\<Union>_" [900] 900)
```
```  1080
```
```  1081 lemma Union_eq:
```
```  1082   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
```
```  1083 proof (rule set_eqI)
```
```  1084   fix x
```
```  1085   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
```
```  1086     by auto
```
```  1087   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
```
```  1088     by (simp add: Sup_set_def image_def)
```
```  1089 qed
```
```  1090
```
```  1091 lemma Union_iff [simp]:
```
```  1092   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
```
```  1093   by (unfold Union_eq) blast
```
```  1094
```
```  1095 lemma UnionI [intro]:
```
```  1096   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
```
```  1097   -- {* The order of the premises presupposes that @{term C} is rigid;
```
```  1098     @{term A} may be flexible. *}
```
```  1099   by auto
```
```  1100
```
```  1101 lemma UnionE [elim!]:
```
```  1102   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
```
```  1103   by auto
```
```  1104
```
```  1105 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
```
```  1106   by (fact Sup_upper)
```
```  1107
```
```  1108 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
```
```  1109   by (fact Sup_least)
```
```  1110
```
```  1111 lemma Union_empty: "\<Union>{} = {}"
```
```  1112   by (fact Sup_empty) (* already simp *)
```
```  1113
```
```  1114 lemma Union_UNIV: "\<Union>UNIV = UNIV"
```
```  1115   by (fact Sup_UNIV) (* already simp *)
```
```  1116
```
```  1117 lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
```
```  1118   by (fact Sup_insert) (* already simp *)
```
```  1119
```
```  1120 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
```
```  1121   by (fact Sup_union_distrib)
```
```  1122
```
```  1123 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
```
```  1124   by (fact Sup_inter_less_eq)
```
```  1125
```
```  1126 lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1127   by (fact Sup_bot_conv) (* already simp *)
```
```  1128
```
```  1129 lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1130   by (fact Sup_bot_conv) (* already simp *)
```
```  1131
```
```  1132 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
```
```  1133   by blast
```
```  1134
```
```  1135 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
```
```  1136   by blast
```
```  1137
```
```  1138 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
```
```  1139   by (fact Sup_subset_mono)
```
```  1140
```
```  1141
```
```  1142 subsubsection {* Unions of families *}
```
```  1143
```
```  1144 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```  1145   "UNION \<equiv> SUPREMUM"
```
```  1146
```
```  1147 text {*
```
```  1148   Note: must use name @{const UNION} here instead of @{text UN}
```
```  1149   to allow the following syntax coexist with the plain constant name.
```
```  1150 *}
```
```  1151
```
```  1152 syntax
```
```  1153   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
```
```  1154   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
```
```  1155
```
```  1156 syntax (xsymbols)
```
```  1157   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
```
```  1158   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1159
```
```  1160 syntax (latex output)
```
```  1161   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```  1162   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```  1163
```
```  1164 translations
```
```  1165   "UN x y. B"   == "UN x. UN y. B"
```
```  1166   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
```
```  1167   "UN x. B"     == "UN x:CONST UNIV. B"
```
```  1168   "UN x:A. B"   == "CONST UNION A (%x. B)"
```
```  1169
```
```  1170 text {*
```
```  1171   Note the difference between ordinary xsymbol syntax of indexed
```
```  1172   unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
```
```  1173   and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
```
```  1174   former does not make the index expression a subscript of the
```
```  1175   union/intersection symbol because this leads to problems with nested
```
```  1176   subscripts in Proof General.
```
```  1177 *}
```
```  1178
```
```  1179 print_translation {*
```
```  1180   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
```
```  1181 *} -- {* to avoid eta-contraction of body *}
```
```  1182
```
```  1183 lemma UNION_eq:
```
```  1184   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
```
```  1185   by (auto intro!: SUP_eqI)
```
```  1186
```
```  1187 lemma bind_UNION [code]:
```
```  1188   "Set.bind A f = UNION A f"
```
```  1189   by (simp add: bind_def UNION_eq)
```
```  1190
```
```  1191 lemma member_bind [simp]:
```
```  1192   "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
```
```  1193   by (simp add: bind_UNION)
```
```  1194
```
```  1195 lemma Union_image_eq:
```
```  1196   "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
```
```  1197   by (fact Sup_image_eq)
```
```  1198
```
```  1199 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
```
```  1200   using Union_iff [of _ "B ` A"] by simp
```
```  1201
```
```  1202 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
```
```  1203   -- {* The order of the premises presupposes that @{term A} is rigid;
```
```  1204     @{term b} may be flexible. *}
```
```  1205   by auto
```
```  1206
```
```  1207 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"
```
```  1208   by (auto simp add: SUP_def image_def)
```
```  1209
```
```  1210 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
```
```  1211   by blast
```
```  1212
```
```  1213 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
```
```  1214   by (fact SUP_upper)
```
```  1215
```
```  1216 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
```
```  1217   by (fact SUP_least)
```
```  1218
```
```  1219 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
```
```  1220   by blast
```
```  1221
```
```  1222 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
```
```  1223   by blast
```
```  1224
```
```  1225 lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
```
```  1226   by (fact SUP_empty)
```
```  1227
```
```  1228 lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
```
```  1229   by (fact SUP_bot) (* already simp *)
```
```  1230
```
```  1231 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
```
```  1232   by (fact SUP_absorb)
```
```  1233
```
```  1234 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
```
```  1235   by (fact SUP_insert)
```
```  1236
```
```  1237 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)"
```
```  1238   by (fact SUP_union)
```
```  1239
```
```  1240 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)"
```
```  1241   by blast
```
```  1242
```
```  1243 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
```
```  1244   by (fact SUP_le_iff)
```
```  1245
```
```  1246 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
```
```  1247   by (fact SUP_constant)
```
```  1248
```
```  1249 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
```
```  1250   by blast
```
```  1251
```
```  1252 lemma UNION_empty_conv:
```
```  1253   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1254   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1255   by (fact SUP_bot_conv)+ (* already simp *)
```
```  1256
```
```  1257 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
```
```  1258   by blast
```
```  1259
```
```  1260 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
```
```  1261   by blast
```
```  1262
```
```  1263 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
```
```  1264   by blast
```
```  1265
```
```  1266 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
```
```  1267   by (auto simp add: split_if_mem2)
```
```  1268
```
```  1269 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
```
```  1270   by (fact SUP_UNIV_bool_expand)
```
```  1271
```
```  1272 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
```
```  1273   by blast
```
```  1274
```
```  1275 lemma UN_mono:
```
```  1276   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
```
```  1277     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
```
```  1278   by (fact SUP_subset_mono)
```
```  1279
```
```  1280 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
```
```  1281   by blast
```
```  1282
```
```  1283 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
```
```  1284   by blast
```
```  1285
```
```  1286 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
```
```  1287   -- {* NOT suitable for rewriting *}
```
```  1288   by blast
```
```  1289
```
```  1290 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
```
```  1291   by blast
```
```  1292
```
```  1293 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
```
```  1294   by blast
```
```  1295
```
```  1296
```
```  1297 subsubsection {* Distributive laws *}
```
```  1298
```
```  1299 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
```
```  1300   by (fact inf_Sup)
```
```  1301
```
```  1302 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
```
```  1303   by (fact sup_Inf)
```
```  1304
```
```  1305 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
```
```  1306   by (fact Sup_inf)
```
```  1307
```
```  1308 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)"
```
```  1309   by (rule sym) (rule INF_inf_distrib)
```
```  1310
```
```  1311 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)"
```
```  1312   by (rule sym) (rule SUP_sup_distrib)
```
```  1313
```
```  1314 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" -- {* FIXME drop *}
```
```  1315   by (simp add: INT_Int_distrib)
```
```  1316
```
```  1317 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" -- {* FIXME drop *}
```
```  1318   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
```
```  1319   -- {* Union of a family of unions *}
```
```  1320   by (simp add: UN_Un_distrib)
```
```  1321
```
```  1322 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
```
```  1323   by (fact sup_INF)
```
```  1324
```
```  1325 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
```
```  1326   -- {* Halmos, Naive Set Theory, page 35. *}
```
```  1327   by (fact inf_SUP)
```
```  1328
```
```  1329 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)"
```
```  1330   by (fact SUP_inf_distrib2)
```
```  1331
```
```  1332 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)"
```
```  1333   by (fact INF_sup_distrib2)
```
```  1334
```
```  1335 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
```
```  1336   by (fact Sup_inf_eq_bot_iff)
```
```  1337
```
```  1338
```
```  1339 subsection {* Injections and bijections *}
```
```  1340
```
```  1341 lemma inj_on_Inter:
```
```  1342   "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
```
```  1343   unfolding inj_on_def by blast
```
```  1344
```
```  1345 lemma inj_on_INTER:
```
```  1346   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)"
```
```  1347   unfolding inj_on_def by blast
```
```  1348
```
```  1349 lemma inj_on_UNION_chain:
```
```  1350   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
```
```  1351          INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1352   shows "inj_on f (\<Union> i \<in> I. A i)"
```
```  1353 proof -
```
```  1354   {
```
```  1355     fix i j x y
```
```  1356     assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
```
```  1357       and ***: "f x = f y"
```
```  1358     have "x = y"
```
```  1359     proof -
```
```  1360       {
```
```  1361         assume "A i \<le> A j"
```
```  1362         with ** have "x \<in> A j" by auto
```
```  1363         with INJ * ** *** have ?thesis
```
```  1364         by(auto simp add: inj_on_def)
```
```  1365       }
```
```  1366       moreover
```
```  1367       {
```
```  1368         assume "A j \<le> A i"
```
```  1369         with ** have "y \<in> A i" by auto
```
```  1370         with INJ * ** *** have ?thesis
```
```  1371         by(auto simp add: inj_on_def)
```
```  1372       }
```
```  1373       ultimately show ?thesis using CH * by blast
```
```  1374     qed
```
```  1375   }
```
```  1376   then show ?thesis by (unfold inj_on_def UNION_eq) auto
```
```  1377 qed
```
```  1378
```
```  1379 lemma bij_betw_UNION_chain:
```
```  1380   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
```
```  1381          BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
```
```  1382   shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
```
```  1383 proof (unfold bij_betw_def, auto)
```
```  1384   have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1385   using BIJ bij_betw_def[of f] by auto
```
```  1386   thus "inj_on f (\<Union> i \<in> I. A i)"
```
```  1387   using CH inj_on_UNION_chain[of I A f] by auto
```
```  1388 next
```
```  1389   fix i x
```
```  1390   assume *: "i \<in> I" "x \<in> A i"
```
```  1391   hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
```
```  1392   thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
```
```  1393 next
```
```  1394   fix i x'
```
```  1395   assume *: "i \<in> I" "x' \<in> A' i"
```
```  1396   hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
```
```  1397   then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
```
```  1398     using * by blast
```
```  1399   then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
```
```  1400 qed
```
```  1401
```
```  1402 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```  1403 lemma image_INT:
```
```  1404    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```  1405     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1406 apply (simp add: inj_on_def, blast)
```
```  1407 done
```
```  1408
```
```  1409 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```  1410   it doesn't matter whether A is empty*)
```
```  1411 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1412 apply (simp add: bij_def)
```
```  1413 apply (simp add: inj_on_def surj_def, blast)
```
```  1414 done
```
```  1415
```
```  1416 lemma UNION_fun_upd:
```
```  1417   "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
```
```  1418 by (auto split: if_splits)
```
```  1419
```
```  1420
```
```  1421 subsubsection {* Complement *}
```
```  1422
```
```  1423 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
```
```  1424   by (fact uminus_INF)
```
```  1425
```
```  1426 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
```
```  1427   by (fact uminus_SUP)
```
```  1428
```
```  1429
```
```  1430 subsubsection {* Miniscoping and maxiscoping *}
```
```  1431
```
```  1432 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
```
```  1433            and Intersections. *}
```
```  1434
```
```  1435 lemma UN_simps [simp]:
```
```  1436   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
```
```  1437   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
```
```  1438   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
```
```  1439   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
```
```  1440   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
```
```  1441   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
```
```  1442   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
```
```  1443   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
```
```  1444   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
```
```  1445   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
```
```  1446   by auto
```
```  1447
```
```  1448 lemma INT_simps [simp]:
```
```  1449   "\<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)"
```
```  1450   "\<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))"
```
```  1451   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
```
```  1452   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
```
```  1453   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
```
```  1454   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
```
```  1455   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
```
```  1456   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
```
```  1457   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
```
```  1458   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
```
```  1459   by auto
```
```  1460
```
```  1461 lemma UN_ball_bex_simps [simp]:
```
```  1462   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
```
```  1463   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
```
```  1464   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
```
```  1465   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
```
```  1466   by auto
```
```  1467
```
```  1468
```
```  1469 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
```
```  1470
```
```  1471 lemma UN_extend_simps:
```
```  1472   "\<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)))"
```
```  1473   "\<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))"
```
```  1474   "\<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))"
```
```  1475   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
```
```  1476   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
```
```  1477   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
```
```  1478   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
```
```  1479   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
```
```  1480   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
```
```  1481   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
```
```  1482   by auto
```
```  1483
```
```  1484 lemma INT_extend_simps:
```
```  1485   "\<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))"
```
```  1486   "\<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))"
```
```  1487   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
```
```  1488   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
```
```  1489   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
```
```  1490   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
```
```  1491   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
```
```  1492   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
```
```  1493   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
```
```  1494   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
```
```  1495   by auto
```
```  1496
```
```  1497 text {* Finally *}
```
```  1498
```
```  1499 no_notation
```
```  1500   less_eq (infix "\<sqsubseteq>" 50) and
```
```  1501   less (infix "\<sqsubset>" 50)
```
```  1502
```
```  1503 lemmas mem_simps =
```
```  1504   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
```
```  1505   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
```
```  1506   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
```
```  1507
```
```  1508 end
```
```  1509
```