src/HOL/Complete_Lattices.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 56076 e52fc7c37ed3 child 56212 3253aaf73a01 permissions -rw-r--r--
normalising simp rules for compound operators
```     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 INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    21   INF_def: "INFI A f = \<Sqinter>(f ` A)"
```
```    22
```
```    23 lemma Inf_image_eq [simp]:
```
```    24   "\<Sqinter>(f ` A) = INFI A f"
```
```    25   by (simp add: INF_def)
```
```    26
```
```    27 lemma INF_image [simp]:
```
```    28   "INFI (f ` A) g = INFI A (g \<circ> f)"
```
```    29   by (simp only: INF_def image_comp)
```
```    30
```
```    31 lemma INF_identity_eq [simp]:
```
```    32   "INFI A (\<lambda>x. x) = \<Sqinter>A"
```
```    33   by (simp add: INF_def)
```
```    34
```
```    35 lemma INF_id_eq [simp]:
```
```    36   "INFI 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> INFI A C = INFI B D"
```
```    41   by (simp add: INF_def image_def)
```
```    42
```
```    43 end
```
```    44
```
```    45 class Sup =
```
```    46   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```    47 begin
```
```    48
```
```    49 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    50   SUP_def: "SUPR A f = \<Squnion>(f ` A)"
```
```    51
```
```    52 lemma Sup_image_eq [simp]:
```
```    53   "\<Squnion>(f ` A) = SUPR A f"
```
```    54   by (simp add: SUP_def)
```
```    55
```
```    56 lemma SUP_image [simp]:
```
```    57   "SUPR (f ` A) g = SUPR A (g \<circ> f)"
```
```    58   by (simp only: SUP_def image_comp)
```
```    59
```
```    60 lemma SUP_identity_eq [simp]:
```
```    61   "SUPR A (\<lambda>x. x) = \<Squnion>A"
```
```    62   by (simp add: SUP_def)
```
```    63
```
```    64 lemma SUP_id_eq [simp]:
```
```    65   "SUPR A id = \<Squnion>A"
```
```    66   by (simp add: id_def)
```
```    67
```
```    68 lemma SUP_cong:
```
```    69   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPR A C = SUPR B D"
```
```    70   by (simp add: SUP_def image_def)
```
```    71
```
```    72 end
```
```    73
```
```    74 text {*
```
```    75   Note: must use names @{const INFI} and @{const SUPR} here instead of
```
```    76   @{text INF} and @{text SUP} to allow the following syntax coexist
```
```    77   with the plain constant names.
```
```    78 *}
```
```    79
```
```    80 syntax
```
```    81   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```    82   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
```
```    83   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```    84   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
```
```    85
```
```    86 syntax (xsymbols)
```
```    87   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
```
```    88   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
```
```    89   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
```
```    90   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
```
```    91
```
```    92 translations
```
```    93   "INF x y. B"   == "INF x. INF y. B"
```
```    94   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
```
```    95   "INF x. B"     == "INF x:CONST UNIV. B"
```
```    96   "INF x:A. B"   == "CONST INFI A (%x. B)"
```
```    97   "SUP x y. B"   == "SUP x. SUP y. B"
```
```    98   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
```
```    99   "SUP x. B"     == "SUP x:CONST UNIV. B"
```
```   100   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
```
```   101
```
```   102 print_translation {*
```
```   103   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
```
```   104     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
```
```   105 *} -- {* to avoid eta-contraction of body *}
```
```   106
```
```   107 subsection {* Abstract complete lattices *}
```
```   108
```
```   109 text {* A complete lattice always has a bottom and a top,
```
```   110 so we include them into the following type class,
```
```   111 along with assumptions that define bottom and top
```
```   112 in terms of infimum and supremum. *}
```
```   113
```
```   114 class complete_lattice = lattice + Inf + Sup + bot + top +
```
```   115   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```   116      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```   117   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```   118      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```   119   assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
```
```   120   assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
```
```   121 begin
```
```   122
```
```   123 subclass bounded_lattice
```
```   124 proof
```
```   125   fix a
```
```   126   show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
```
```   127   show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
```
```   128 qed
```
```   129
```
```   130 lemma dual_complete_lattice:
```
```   131   "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   132   by (auto intro!: class.complete_lattice.intro dual_lattice)
```
```   133     (unfold_locales, (fact Inf_empty Sup_empty
```
```   134         Sup_upper Sup_least Inf_lower Inf_greatest)+)
```
```   135
```
```   136 end
```
```   137
```
```   138 context complete_lattice
```
```   139 begin
```
```   140
```
```   141 lemma INF_foundation_dual:
```
```   142   "Sup.SUPR Inf = INFI"
```
```   143   by (simp add: fun_eq_iff Sup.SUP_def)
```
```   144
```
```   145 lemma SUP_foundation_dual:
```
```   146   "Inf.INFI Sup = SUPR"
```
```   147   by (simp add: fun_eq_iff Inf.INF_def)
```
```   148
```
```   149 lemma Sup_eqI:
```
```   150   "(\<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"
```
```   151   by (blast intro: antisym Sup_least Sup_upper)
```
```   152
```
```   153 lemma Inf_eqI:
```
```   154   "(\<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"
```
```   155   by (blast intro: antisym Inf_greatest Inf_lower)
```
```   156
```
```   157 lemma SUP_eqI:
```
```   158   "(\<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"
```
```   159   using Sup_eqI [of "f ` A" x] by auto
```
```   160
```
```   161 lemma INF_eqI:
```
```   162   "(\<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"
```
```   163   using Inf_eqI [of "f ` A" x] by auto
```
```   164
```
```   165 lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
```
```   166   using Inf_lower [of _ "f ` A"] by simp
```
```   167
```
```   168 lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
```
```   169   using Inf_greatest [of "f ` A"] by auto
```
```   170
```
```   171 lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   172   using Sup_upper [of _ "f ` A"] by simp
```
```   173
```
```   174 lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
```
```   175   using Sup_least [of "f ` A"] by auto
```
```   176
```
```   177 lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
```
```   178   using Inf_lower [of u A] by auto
```
```   179
```
```   180 lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
```
```   181   using INF_lower [of i A f] by auto
```
```   182
```
```   183 lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
```
```   184   using Sup_upper [of u A] by auto
```
```   185
```
```   186 lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
```
```   187   using SUP_upper [of i A f] by auto
```
```   188
```
```   189 lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
```
```   190   by (auto intro: Inf_greatest dest: Inf_lower)
```
```   191
```
```   192 lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
```
```   193   using le_Inf_iff [of _ "f ` A"] by simp
```
```   194
```
```   195 lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
```
```   196   by (auto intro: Sup_least dest: Sup_upper)
```
```   197
```
```   198 lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
```
```   199   using Sup_le_iff [of "f ` A"] by simp
```
```   200
```
```   201 lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   202   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
```
```   203
```
```   204 lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
```
```   205   unfolding INF_def Inf_insert by simp
```
```   206
```
```   207 lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   208   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
```
```   209
```
```   210 lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
```
```   211   unfolding SUP_def Sup_insert by simp
```
```   212
```
```   213 lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
```
```   214   by (simp add: INF_def)
```
```   215
```
```   216 lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
```
```   217   by (simp add: SUP_def)
```
```   218
```
```   219 lemma Inf_UNIV [simp]:
```
```   220   "\<Sqinter>UNIV = \<bottom>"
```
```   221   by (auto intro!: antisym Inf_lower)
```
```   222
```
```   223 lemma Sup_UNIV [simp]:
```
```   224   "\<Squnion>UNIV = \<top>"
```
```   225   by (auto intro!: antisym Sup_upper)
```
```   226
```
```   227 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
```
```   228   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   229
```
```   230 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
```
```   231   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   232
```
```   233 lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   234   by (auto intro: Inf_greatest Inf_lower)
```
```   235
```
```   236 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   237   by (auto intro: Sup_least Sup_upper)
```
```   238
```
```   239 lemma Inf_mono:
```
```   240   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
```
```   241   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
```
```   242 proof (rule Inf_greatest)
```
```   243   fix b assume "b \<in> B"
```
```   244   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
```
```   245   from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
```
```   246   with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
```
```   247 qed
```
```   248
```
```   249 lemma INF_mono:
```
```   250   "(\<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)"
```
```   251   using Inf_mono [of "g ` B" "f ` A"] by auto
```
```   252
```
```   253 lemma Sup_mono:
```
```   254   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
```
```   255   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
```
```   256 proof (rule Sup_least)
```
```   257   fix a assume "a \<in> A"
```
```   258   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
```
```   259   from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
```
```   260   with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
```
```   261 qed
```
```   262
```
```   263 lemma SUP_mono:
```
```   264   "(\<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)"
```
```   265   using Sup_mono [of "f ` A" "g ` B"] by auto
```
```   266
```
```   267 lemma INF_superset_mono:
```
```   268   "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)"
```
```   269   -- {* The last inclusion is POSITIVE! *}
```
```   270   by (blast intro: INF_mono dest: subsetD)
```
```   271
```
```   272 lemma SUP_subset_mono:
```
```   273   "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)"
```
```   274   by (blast intro: SUP_mono dest: subsetD)
```
```   275
```
```   276 lemma Inf_less_eq:
```
```   277   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
```
```   278     and "A \<noteq> {}"
```
```   279   shows "\<Sqinter>A \<sqsubseteq> u"
```
```   280 proof -
```
```   281   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   282   moreover from `v \<in> A` assms(1) have "v \<sqsubseteq> u" by blast
```
```   283   ultimately show ?thesis by (rule Inf_lower2)
```
```   284 qed
```
```   285
```
```   286 lemma less_eq_Sup:
```
```   287   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
```
```   288     and "A \<noteq> {}"
```
```   289   shows "u \<sqsubseteq> \<Squnion>A"
```
```   290 proof -
```
```   291   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
```
```   292   moreover from `v \<in> A` assms(1) have "u \<sqsubseteq> v" by blast
```
```   293   ultimately show ?thesis by (rule Sup_upper2)
```
```   294 qed
```
```   295
```
```   296 lemma SUPR_eq:
```
```   297   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
```
```   298   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
```
```   299   shows "(\<Squnion>i\<in>A. f i) = (\<Squnion>j\<in>B. g j)"
```
```   300   by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
```
```   301
```
```   302 lemma INFI_eq:
```
```   303   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
```
```   304   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
```
```   305   shows "(\<Sqinter>i\<in>A. f i) = (\<Sqinter>j\<in>B. g j)"
```
```   306   by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
```
```   307
```
```   308 lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
```
```   309   by (auto intro: Inf_greatest Inf_lower)
```
```   310
```
```   311 lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
```
```   312   by (auto intro: Sup_least Sup_upper)
```
```   313
```
```   314 lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
```
```   315   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
```
```   316
```
```   317 lemma INF_union:
```
```   318   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
```
```   319   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
```
```   320
```
```   321 lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
```
```   322   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
```
```   323
```
```   324 lemma SUP_union:
```
```   325   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
```
```   326   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
```
```   327
```
```   328 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)"
```
```   329   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
```
```   330
```
```   331 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")
```
```   332 proof (rule antisym)
```
```   333   show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
```
```   334 next
```
```   335   show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
```
```   336 qed
```
```   337
```
```   338 lemma Inf_top_conv [simp]:
```
```   339   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   340   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   341 proof -
```
```   342   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
```
```   343   proof
```
```   344     assume "\<forall>x\<in>A. x = \<top>"
```
```   345     then have "A = {} \<or> A = {\<top>}" by auto
```
```   346     then show "\<Sqinter>A = \<top>" by auto
```
```   347   next
```
```   348     assume "\<Sqinter>A = \<top>"
```
```   349     show "\<forall>x\<in>A. x = \<top>"
```
```   350     proof (rule ccontr)
```
```   351       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
```
```   352       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
```
```   353       then obtain B where "A = insert x B" by blast
```
```   354       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
```
```   355     qed
```
```   356   qed
```
```   357   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
```
```   358 qed
```
```   359
```
```   360 lemma INF_top_conv [simp]:
```
```   361   "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   362   "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
```
```   363   using Inf_top_conv [of "B ` A"] by simp_all
```
```   364
```
```   365 lemma Sup_bot_conv [simp]:
```
```   366   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
```
```   367   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
```
```   368   using dual_complete_lattice
```
```   369   by (rule complete_lattice.Inf_top_conv)+
```
```   370
```
```   371 lemma SUP_bot_conv [simp]:
```
```   372  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   373  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
```
```   374   using Sup_bot_conv [of "B ` A"] by simp_all
```
```   375
```
```   376 lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
```
```   377   by (auto intro: antisym INF_lower INF_greatest)
```
```   378
```
```   379 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
```
```   380   by (auto intro: antisym SUP_upper SUP_least)
```
```   381
```
```   382 lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
```
```   383   by (cases "A = {}") simp_all
```
```   384
```
```   385 lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
```
```   386   by (cases "A = {}") simp_all
```
```   387
```
```   388 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)"
```
```   389   by (iprover intro: INF_lower INF_greatest order_trans antisym)
```
```   390
```
```   391 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)"
```
```   392   by (iprover intro: SUP_upper SUP_least order_trans antisym)
```
```   393
```
```   394 lemma INF_absorb:
```
```   395   assumes "k \<in> I"
```
```   396   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
```
```   397 proof -
```
```   398   from assms obtain J where "I = insert k J" by blast
```
```   399   then show ?thesis by simp
```
```   400 qed
```
```   401
```
```   402 lemma SUP_absorb:
```
```   403   assumes "k \<in> I"
```
```   404   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>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 INF_constant:
```
```   411   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
```
```   412   by simp
```
```   413
```
```   414 lemma SUP_constant:
```
```   415   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
```
```   416   by simp
```
```   417
```
```   418 lemma less_INF_D:
```
```   419   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
```
```   420 proof -
```
```   421   note `y < (\<Sqinter>i\<in>A. f i)`
```
```   422   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
```
```   423     by (rule INF_lower)
```
```   424   finally show "y < f i" .
```
```   425 qed
```
```   426
```
```   427 lemma SUP_lessD:
```
```   428   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
```
```   429 proof -
```
```   430   have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
```
```   431     by (rule SUP_upper)
```
```   432   also note `(\<Squnion>i\<in>A. f i) < y`
```
```   433   finally show "f i < y" .
```
```   434 qed
```
```   435
```
```   436 lemma INF_UNIV_bool_expand:
```
```   437   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
```
```   438   by (simp add: UNIV_bool inf_commute)
```
```   439
```
```   440 lemma SUP_UNIV_bool_expand:
```
```   441   "(\<Squnion>b. A b) = A True \<squnion> A False"
```
```   442   by (simp add: UNIV_bool sup_commute)
```
```   443
```
```   444 lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
```
```   445   by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
```
```   446
```
```   447 lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f"
```
```   448   using Inf_le_Sup [of "f ` A"] by simp
```
```   449
```
```   450 lemma SUP_eq_const:
```
```   451   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> SUPR I f = x"
```
```   452   by (auto intro: SUP_eqI)
```
```   453
```
```   454 lemma INF_eq_const:
```
```   455   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> INFI I f = x"
```
```   456   by (auto intro: INF_eqI)
```
```   457
```
```   458 lemma SUP_eq_iff:
```
```   459   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> (SUPR I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   460   using SUP_eq_const[of I f c] SUP_upper[of _ I f] by (auto intro: antisym)
```
```   461
```
```   462 lemma INF_eq_iff:
```
```   463   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> (INFI I f = c) \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
```
```   464   using INF_eq_const[of I f c] INF_lower[of _ I f] by (auto intro: antisym)
```
```   465
```
```   466 end
```
```   467
```
```   468 class complete_distrib_lattice = complete_lattice +
```
```   469   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
```
```   470   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   471 begin
```
```   472
```
```   473 lemma sup_INF:
```
```   474   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
```
```   475   by (simp only: INF_def sup_Inf image_image)
```
```   476
```
```   477 lemma inf_SUP:
```
```   478   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
```
```   479   by (simp only: SUP_def inf_Sup image_image)
```
```   480
```
```   481 lemma dual_complete_distrib_lattice:
```
```   482   "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   483   apply (rule class.complete_distrib_lattice.intro)
```
```   484   apply (fact dual_complete_lattice)
```
```   485   apply (rule class.complete_distrib_lattice_axioms.intro)
```
```   486   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
```
```   487   done
```
```   488
```
```   489 subclass distrib_lattice proof
```
```   490   fix a b c
```
```   491   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
```
```   492   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
```
```   493 qed
```
```   494
```
```   495 lemma Inf_sup:
```
```   496   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
```
```   497   by (simp add: sup_Inf sup_commute)
```
```   498
```
```   499 lemma Sup_inf:
```
```   500   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
```
```   501   by (simp add: inf_Sup inf_commute)
```
```   502
```
```   503 lemma INF_sup:
```
```   504   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
```
```   505   by (simp add: sup_INF sup_commute)
```
```   506
```
```   507 lemma SUP_inf:
```
```   508   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
```
```   509   by (simp add: inf_SUP inf_commute)
```
```   510
```
```   511 lemma Inf_sup_eq_top_iff:
```
```   512   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
```
```   513   by (simp only: Inf_sup INF_top_conv)
```
```   514
```
```   515 lemma Sup_inf_eq_bot_iff:
```
```   516   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
```
```   517   by (simp only: Sup_inf SUP_bot_conv)
```
```   518
```
```   519 lemma INF_sup_distrib2:
```
```   520   "(\<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)"
```
```   521   by (subst INF_commute) (simp add: sup_INF INF_sup)
```
```   522
```
```   523 lemma SUP_inf_distrib2:
```
```   524   "(\<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)"
```
```   525   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
```
```   526
```
```   527 context
```
```   528   fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
```
```   529   assumes "mono f"
```
```   530 begin
```
```   531
```
```   532 lemma mono_Inf:
```
```   533   shows "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
```
```   534   using `mono f` by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)
```
```   535
```
```   536 lemma mono_Sup:
```
```   537   shows "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
```
```   538   using `mono f` by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)
```
```   539
```
```   540 end
```
```   541
```
```   542 end
```
```   543
```
```   544 class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
```
```   545 begin
```
```   546
```
```   547 lemma dual_complete_boolean_algebra:
```
```   548   "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
```
```   549   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
```
```   550
```
```   551 lemma uminus_Inf:
```
```   552   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
```
```   553 proof (rule antisym)
```
```   554   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
```
```   555     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
```
```   556   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
```
```   557     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
```
```   558 qed
```
```   559
```
```   560 lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
```
```   561   by (simp only: INF_def SUP_def uminus_Inf image_image)
```
```   562
```
```   563 lemma uminus_Sup:
```
```   564   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
```
```   565 proof -
```
```   566   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_INF)
```
```   567   then show ?thesis by simp
```
```   568 qed
```
```   569
```
```   570 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
```
```   571   by (simp only: INF_def SUP_def uminus_Sup image_image)
```
```   572
```
```   573 end
```
```   574
```
```   575 class complete_linorder = linorder + complete_lattice
```
```   576 begin
```
```   577
```
```   578 lemma dual_complete_linorder:
```
```   579   "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
```
```   580   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
```
```   581
```
```   582 lemma complete_linorder_inf_min: "inf = min"
```
```   583   by (auto intro: antisym simp add: min_def fun_eq_iff)
```
```   584
```
```   585 lemma complete_linorder_sup_max: "sup = max"
```
```   586   by (auto intro: antisym simp add: max_def fun_eq_iff)
```
```   587
```
```   588 lemma Inf_less_iff:
```
```   589   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
```
```   590   unfolding not_le [symmetric] le_Inf_iff by auto
```
```   591
```
```   592 lemma INF_less_iff:
```
```   593   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
```
```   594   using Inf_less_iff [of "f ` A"] by simp
```
```   595
```
```   596 lemma less_Sup_iff:
```
```   597   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
```
```   598   unfolding not_le [symmetric] Sup_le_iff by auto
```
```   599
```
```   600 lemma less_SUP_iff:
```
```   601   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
```
```   602   using less_Sup_iff [of _ "f ` A"] by simp
```
```   603
```
```   604 lemma Sup_eq_top_iff [simp]:
```
```   605   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
```
```   606 proof
```
```   607   assume *: "\<Squnion>A = \<top>"
```
```   608   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
```
```   609   proof (intro allI impI)
```
```   610     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
```
```   611       unfolding less_Sup_iff by auto
```
```   612   qed
```
```   613 next
```
```   614   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
```
```   615   show "\<Squnion>A = \<top>"
```
```   616   proof (rule ccontr)
```
```   617     assume "\<Squnion>A \<noteq> \<top>"
```
```   618     with top_greatest [of "\<Squnion>A"]
```
```   619     have "\<Squnion>A < \<top>" unfolding le_less by auto
```
```   620     then have "\<Squnion>A < \<Squnion>A"
```
```   621       using * unfolding less_Sup_iff by auto
```
```   622     then show False by auto
```
```   623   qed
```
```   624 qed
```
```   625
```
```   626 lemma SUP_eq_top_iff [simp]:
```
```   627   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
```
```   628   using Sup_eq_top_iff [of "f ` A"] by simp
```
```   629
```
```   630 lemma Inf_eq_bot_iff [simp]:
```
```   631   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
```
```   632   using dual_complete_linorder
```
```   633   by (rule complete_linorder.Sup_eq_top_iff)
```
```   634
```
```   635 lemma INF_eq_bot_iff [simp]:
```
```   636   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
```
```   637   using Inf_eq_bot_iff [of "f ` A"] by simp
```
```   638
```
```   639 lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
```
```   640 proof safe
```
```   641   fix y assume "x \<ge> \<Sqinter>A" "y > x"
```
```   642   then have "y > \<Sqinter>A" by auto
```
```   643   then show "\<exists>a\<in>A. y > a"
```
```   644     unfolding Inf_less_iff .
```
```   645 qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
```
```   646
```
```   647 lemma INF_le_iff:
```
```   648   "INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
```
```   649   using Inf_le_iff [of "f ` A"] by simp
```
```   650
```
```   651 lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
```
```   652 proof safe
```
```   653   fix y assume "x \<le> \<Squnion>A" "y < x"
```
```   654   then have "y < \<Squnion>A" by auto
```
```   655   then show "\<exists>a\<in>A. y < a"
```
```   656     unfolding less_Sup_iff .
```
```   657 qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
```
```   658
```
```   659 lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
```
```   660   using le_Sup_iff [of _ "f ` A"] by simp
```
```   661
```
```   662 subclass complete_distrib_lattice
```
```   663 proof
```
```   664   fix a and B
```
```   665   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
```
```   666     by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
```
```   667       (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
```
```   668         le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
```
```   669 qed
```
```   670
```
```   671 end
```
```   672
```
```   673
```
```   674 subsection {* Complete lattice on @{typ bool} *}
```
```   675
```
```   676 instantiation bool :: complete_lattice
```
```   677 begin
```
```   678
```
```   679 definition
```
```   680   [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
```
```   681
```
```   682 definition
```
```   683   [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
```
```   684
```
```   685 instance proof
```
```   686 qed (auto intro: bool_induct)
```
```   687
```
```   688 end
```
```   689
```
```   690 lemma not_False_in_image_Ball [simp]:
```
```   691   "False \<notin> P ` A \<longleftrightarrow> Ball A P"
```
```   692   by auto
```
```   693
```
```   694 lemma True_in_image_Bex [simp]:
```
```   695   "True \<in> P ` A \<longleftrightarrow> Bex A P"
```
```   696   by auto
```
```   697
```
```   698 lemma INF_bool_eq [simp]:
```
```   699   "INFI = Ball"
```
```   700   by (simp add: fun_eq_iff INF_def)
```
```   701
```
```   702 lemma SUP_bool_eq [simp]:
```
```   703   "SUPR = Bex"
```
```   704   by (simp add: fun_eq_iff SUP_def)
```
```   705
```
```   706 instance bool :: complete_boolean_algebra proof
```
```   707 qed (auto intro: bool_induct)
```
```   708
```
```   709
```
```   710 subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
```
```   711
```
```   712 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   713 begin
```
```   714
```
```   715 definition
```
```   716   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
```
```   717
```
```   718 lemma Inf_apply [simp, code]:
```
```   719   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
```
```   720   by (simp add: Inf_fun_def)
```
```   721
```
```   722 definition
```
```   723   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
```
```   724
```
```   725 lemma Sup_apply [simp, code]:
```
```   726   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
```
```   727   by (simp add: Sup_fun_def)
```
```   728
```
```   729 instance proof
```
```   730 qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
```
```   731
```
```   732 end
```
```   733
```
```   734 lemma INF_apply [simp]:
```
```   735   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
```
```   736   using Inf_apply [of "f ` A"] by (simp add: comp_def)
```
```   737
```
```   738 lemma SUP_apply [simp]:
```
```   739   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
```
```   740   using Sup_apply [of "f ` A"] by (simp add: comp_def)
```
```   741
```
```   742 instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
```
```   743 qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf fun_eq_iff image_image
```
```   744   simp del: Inf_image_eq Sup_image_eq)
```
```   745
```
```   746 instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
```
```   747
```
```   748
```
```   749 subsection {* Complete lattice on unary and binary predicates *}
```
```   750
```
```   751 lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
```
```   752   by simp
```
```   753
```
```   754 lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
```
```   755   by simp
```
```   756
```
```   757 lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
```
```   758   by auto
```
```   759
```
```   760 lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
```
```   761   by auto
```
```   762
```
```   763 lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
```
```   764   by auto
```
```   765
```
```   766 lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
```
```   767   by auto
```
```   768
```
```   769 lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
```
```   770   by auto
```
```   771
```
```   772 lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
```
```   773   by auto
```
```   774
```
```   775 lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
```
```   776   by simp
```
```   777
```
```   778 lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
```
```   779   by simp
```
```   780
```
```   781 lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
```
```   782   by auto
```
```   783
```
```   784 lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
```
```   785   by auto
```
```   786
```
```   787 lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
```
```   788   by auto
```
```   789
```
```   790 lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
```
```   791   by auto
```
```   792
```
```   793
```
```   794 subsection {* Complete lattice on @{typ "_ set"} *}
```
```   795
```
```   796 instantiation "set" :: (type) complete_lattice
```
```   797 begin
```
```   798
```
```   799 definition
```
```   800   "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
```
```   801
```
```   802 definition
```
```   803   "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
```
```   804
```
```   805 instance proof
```
```   806 qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
```
```   807
```
```   808 end
```
```   809
```
```   810 instance "set" :: (type) complete_boolean_algebra
```
```   811 proof
```
```   812 qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
```
```   813
```
```   814
```
```   815 subsubsection {* Inter *}
```
```   816
```
```   817 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
```
```   818   "Inter S \<equiv> \<Sqinter>S"
```
```   819
```
```   820 notation (xsymbols)
```
```   821   Inter  ("\<Inter>_" [900] 900)
```
```   822
```
```   823 lemma Inter_eq:
```
```   824   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
```
```   825 proof (rule set_eqI)
```
```   826   fix x
```
```   827   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
```
```   828     by auto
```
```   829   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
```
```   830     by (simp add: Inf_set_def image_def)
```
```   831 qed
```
```   832
```
```   833 lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
```
```   834   by (unfold Inter_eq) blast
```
```   835
```
```   836 lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
```
```   837   by (simp add: Inter_eq)
```
```   838
```
```   839 text {*
```
```   840   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
```
```   841   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
```
```   842   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
```
```   843 *}
```
```   844
```
```   845 lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
```
```   846   by auto
```
```   847
```
```   848 lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
```
```   849   -- {* ``Classical'' elimination rule -- does not require proving
```
```   850     @{prop "X \<in> C"}. *}
```
```   851   by (unfold Inter_eq) blast
```
```   852
```
```   853 lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   854   by (fact Inf_lower)
```
```   855
```
```   856 lemma Inter_subset:
```
```   857   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
```
```   858   by (fact Inf_less_eq)
```
```   859
```
```   860 lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
```
```   861   by (fact Inf_greatest)
```
```   862
```
```   863 lemma Inter_empty: "\<Inter>{} = UNIV"
```
```   864   by (fact Inf_empty) (* already simp *)
```
```   865
```
```   866 lemma Inter_UNIV: "\<Inter>UNIV = {}"
```
```   867   by (fact Inf_UNIV) (* already simp *)
```
```   868
```
```   869 lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
```
```   870   by (fact Inf_insert) (* already simp *)
```
```   871
```
```   872 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
```
```   873   by (fact less_eq_Inf_inter)
```
```   874
```
```   875 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
```
```   876   by (fact Inf_union_distrib)
```
```   877
```
```   878 lemma Inter_UNIV_conv [simp]:
```
```   879   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   880   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
```
```   881   by (fact Inf_top_conv)+
```
```   882
```
```   883 lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
```
```   884   by (fact Inf_superset_mono)
```
```   885
```
```   886
```
```   887 subsubsection {* Intersections of families *}
```
```   888
```
```   889 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```   890   "INTER \<equiv> INFI"
```
```   891
```
```   892 text {*
```
```   893   Note: must use name @{const INTER} here instead of @{text INT}
```
```   894   to allow the following syntax coexist with the plain constant name.
```
```   895 *}
```
```   896
```
```   897 syntax
```
```   898   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
```
```   899   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
```
```   900
```
```   901 syntax (xsymbols)
```
```   902   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
```
```   903   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
```
```   904
```
```   905 syntax (latex output)
```
```   906   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```   907   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```   908
```
```   909 translations
```
```   910   "INT x y. B"  == "INT x. INT y. B"
```
```   911   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
```
```   912   "INT x. B"    == "INT x:CONST UNIV. B"
```
```   913   "INT x:A. B"  == "CONST INTER A (%x. B)"
```
```   914
```
```   915 print_translation {*
```
```   916   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
```
```   917 *} -- {* to avoid eta-contraction of body *}
```
```   918
```
```   919 lemma INTER_eq:
```
```   920   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
```
```   921   by (auto intro!: INF_eqI)
```
```   922
```
```   923 lemma Inter_image_eq:
```
```   924   "\<Inter>(B ` A) = (\<Inter>x\<in>A. B x)"
```
```   925   by (fact Inf_image_eq)
```
```   926
```
```   927 lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
```
```   928   using Inter_iff [of _ "B ` A"] by simp
```
```   929
```
```   930 lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
```
```   931   by (auto simp add: INF_def image_def)
```
```   932
```
```   933 lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
```
```   934   by auto
```
```   935
```
```   936 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"
```
```   937   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
```
```   938   by (auto simp add: INF_def image_def)
```
```   939
```
```   940 lemma INT_cong [cong]:
```
```   941   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
```
```   942   by (fact INF_cong)
```
```   943
```
```   944 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
```
```   945   by blast
```
```   946
```
```   947 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
```
```   948   by blast
```
```   949
```
```   950 lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
```
```   951   by (fact INF_lower)
```
```   952
```
```   953 lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
```
```   954   by (fact INF_greatest)
```
```   955
```
```   956 lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
```
```   957   by (fact INF_empty)
```
```   958
```
```   959 lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
```
```   960   by (fact INF_absorb)
```
```   961
```
```   962 lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
```
```   963   by (fact le_INF_iff)
```
```   964
```
```   965 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
```
```   966   by (fact INF_insert)
```
```   967
```
```   968 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
```
```   969   by (fact INF_union)
```
```   970
```
```   971 lemma INT_insert_distrib:
```
```   972   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
```
```   973   by blast
```
```   974
```
```   975 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
```
```   976   by (fact INF_constant)
```
```   977
```
```   978 lemma INTER_UNIV_conv:
```
```   979  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
```
```   980  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
```
```   981   by (fact INF_top_conv)+ (* already simp *)
```
```   982
```
```   983 lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
```
```   984   by (fact INF_UNIV_bool_expand)
```
```   985
```
```   986 lemma INT_anti_mono:
```
```   987   "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)"
```
```   988   -- {* The last inclusion is POSITIVE! *}
```
```   989   by (fact INF_superset_mono)
```
```   990
```
```   991 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
```
```   992   by blast
```
```   993
```
```   994 lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
```
```   995   by blast
```
```   996
```
```   997
```
```   998 subsubsection {* Union *}
```
```   999
```
```  1000 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
```
```  1001   "Union S \<equiv> \<Squnion>S"
```
```  1002
```
```  1003 notation (xsymbols)
```
```  1004   Union  ("\<Union>_" [900] 900)
```
```  1005
```
```  1006 lemma Union_eq:
```
```  1007   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
```
```  1008 proof (rule set_eqI)
```
```  1009   fix x
```
```  1010   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
```
```  1011     by auto
```
```  1012   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
```
```  1013     by (simp add: Sup_set_def image_def)
```
```  1014 qed
```
```  1015
```
```  1016 lemma Union_iff [simp]:
```
```  1017   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
```
```  1018   by (unfold Union_eq) blast
```
```  1019
```
```  1020 lemma UnionI [intro]:
```
```  1021   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
```
```  1022   -- {* The order of the premises presupposes that @{term C} is rigid;
```
```  1023     @{term A} may be flexible. *}
```
```  1024   by auto
```
```  1025
```
```  1026 lemma UnionE [elim!]:
```
```  1027   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
```
```  1028   by auto
```
```  1029
```
```  1030 lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
```
```  1031   by (fact Sup_upper)
```
```  1032
```
```  1033 lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
```
```  1034   by (fact Sup_least)
```
```  1035
```
```  1036 lemma Union_empty: "\<Union>{} = {}"
```
```  1037   by (fact Sup_empty) (* already simp *)
```
```  1038
```
```  1039 lemma Union_UNIV: "\<Union>UNIV = UNIV"
```
```  1040   by (fact Sup_UNIV) (* already simp *)
```
```  1041
```
```  1042 lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
```
```  1043   by (fact Sup_insert) (* already simp *)
```
```  1044
```
```  1045 lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
```
```  1046   by (fact Sup_union_distrib)
```
```  1047
```
```  1048 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
```
```  1049   by (fact Sup_inter_less_eq)
```
```  1050
```
```  1051 lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1052   by (fact Sup_bot_conv) (* already simp *)
```
```  1053
```
```  1054 lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
```
```  1055   by (fact Sup_bot_conv) (* already simp *)
```
```  1056
```
```  1057 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
```
```  1058   by blast
```
```  1059
```
```  1060 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
```
```  1061   by blast
```
```  1062
```
```  1063 lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
```
```  1064   by (fact Sup_subset_mono)
```
```  1065
```
```  1066
```
```  1067 subsubsection {* Unions of families *}
```
```  1068
```
```  1069 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
```
```  1070   "UNION \<equiv> SUPR"
```
```  1071
```
```  1072 text {*
```
```  1073   Note: must use name @{const UNION} here instead of @{text UN}
```
```  1074   to allow the following syntax coexist with the plain constant name.
```
```  1075 *}
```
```  1076
```
```  1077 syntax
```
```  1078   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
```
```  1079   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
```
```  1080
```
```  1081 syntax (xsymbols)
```
```  1082   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
```
```  1083   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
```
```  1084
```
```  1085 syntax (latex output)
```
```  1086   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
```
```  1087   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
```
```  1088
```
```  1089 translations
```
```  1090   "UN x y. B"   == "UN x. UN y. B"
```
```  1091   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
```
```  1092   "UN x. B"     == "UN x:CONST UNIV. B"
```
```  1093   "UN x:A. B"   == "CONST UNION A (%x. B)"
```
```  1094
```
```  1095 text {*
```
```  1096   Note the difference between ordinary xsymbol syntax of indexed
```
```  1097   unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
```
```  1098   and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
```
```  1099   former does not make the index expression a subscript of the
```
```  1100   union/intersection symbol because this leads to problems with nested
```
```  1101   subscripts in Proof General.
```
```  1102 *}
```
```  1103
```
```  1104 print_translation {*
```
```  1105   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
```
```  1106 *} -- {* to avoid eta-contraction of body *}
```
```  1107
```
```  1108 lemma UNION_eq:
```
```  1109   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
```
```  1110   by (auto intro!: SUP_eqI)
```
```  1111
```
```  1112 lemma bind_UNION [code]:
```
```  1113   "Set.bind A f = UNION A f"
```
```  1114   by (simp add: bind_def UNION_eq)
```
```  1115
```
```  1116 lemma member_bind [simp]:
```
```  1117   "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
```
```  1118   by (simp add: bind_UNION)
```
```  1119
```
```  1120 lemma Union_image_eq:
```
```  1121   "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
```
```  1122   by (fact Sup_image_eq)
```
```  1123
```
```  1124 lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
```
```  1125   using Union_iff [of _ "B ` A"] by simp
```
```  1126
```
```  1127 lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
```
```  1128   -- {* The order of the premises presupposes that @{term A} is rigid;
```
```  1129     @{term b} may be flexible. *}
```
```  1130   by auto
```
```  1131
```
```  1132 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"
```
```  1133   by (auto simp add: SUP_def image_def)
```
```  1134
```
```  1135 lemma UN_cong [cong]:
```
```  1136   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
```
```  1137   by (fact SUP_cong)
```
```  1138
```
```  1139 lemma strong_UN_cong:
```
```  1140   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
```
```  1141   by (unfold simp_implies_def) (fact UN_cong)
```
```  1142
```
```  1143 lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
```
```  1144   by blast
```
```  1145
```
```  1146 lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
```
```  1147   by (fact SUP_upper)
```
```  1148
```
```  1149 lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
```
```  1150   by (fact SUP_least)
```
```  1151
```
```  1152 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
```
```  1153   by blast
```
```  1154
```
```  1155 lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
```
```  1156   by blast
```
```  1157
```
```  1158 lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
```
```  1159   by (fact SUP_empty)
```
```  1160
```
```  1161 lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
```
```  1162   by (fact SUP_bot) (* already simp *)
```
```  1163
```
```  1164 lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
```
```  1165   by (fact SUP_absorb)
```
```  1166
```
```  1167 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
```
```  1168   by (fact SUP_insert)
```
```  1169
```
```  1170 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)"
```
```  1171   by (fact SUP_union)
```
```  1172
```
```  1173 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)"
```
```  1174   by blast
```
```  1175
```
```  1176 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
```
```  1177   by (fact SUP_le_iff)
```
```  1178
```
```  1179 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
```
```  1180   by (fact SUP_constant)
```
```  1181
```
```  1182 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
```
```  1183   by blast
```
```  1184
```
```  1185 lemma UNION_empty_conv:
```
```  1186   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1187   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
```
```  1188   by (fact SUP_bot_conv)+ (* already simp *)
```
```  1189
```
```  1190 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
```
```  1191   by blast
```
```  1192
```
```  1193 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
```
```  1194   by blast
```
```  1195
```
```  1196 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
```
```  1197   by blast
```
```  1198
```
```  1199 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
```
```  1200   by (auto simp add: split_if_mem2)
```
```  1201
```
```  1202 lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
```
```  1203   by (fact SUP_UNIV_bool_expand)
```
```  1204
```
```  1205 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
```
```  1206   by blast
```
```  1207
```
```  1208 lemma UN_mono:
```
```  1209   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
```
```  1210     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
```
```  1211   by (fact SUP_subset_mono)
```
```  1212
```
```  1213 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
```
```  1214   by blast
```
```  1215
```
```  1216 lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
```
```  1217   by blast
```
```  1218
```
```  1219 lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
```
```  1220   -- {* NOT suitable for rewriting *}
```
```  1221   by blast
```
```  1222
```
```  1223 lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
```
```  1224   by blast
```
```  1225
```
```  1226 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
```
```  1227   by blast
```
```  1228
```
```  1229
```
```  1230 subsubsection {* Distributive laws *}
```
```  1231
```
```  1232 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
```
```  1233   by (fact inf_Sup)
```
```  1234
```
```  1235 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
```
```  1236   by (fact sup_Inf)
```
```  1237
```
```  1238 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
```
```  1239   by (fact Sup_inf)
```
```  1240
```
```  1241 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)"
```
```  1242   by (rule sym) (rule INF_inf_distrib)
```
```  1243
```
```  1244 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)"
```
```  1245   by (rule sym) (rule SUP_sup_distrib)
```
```  1246
```
```  1247 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" -- {* FIXME drop *}
```
```  1248   by (simp add: INT_Int_distrib)
```
```  1249
```
```  1250 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" -- {* FIXME drop *}
```
```  1251   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
```
```  1252   -- {* Union of a family of unions *}
```
```  1253   by (simp add: UN_Un_distrib)
```
```  1254
```
```  1255 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
```
```  1256   by (fact sup_INF)
```
```  1257
```
```  1258 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
```
```  1259   -- {* Halmos, Naive Set Theory, page 35. *}
```
```  1260   by (fact inf_SUP)
```
```  1261
```
```  1262 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)"
```
```  1263   by (fact SUP_inf_distrib2)
```
```  1264
```
```  1265 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)"
```
```  1266   by (fact INF_sup_distrib2)
```
```  1267
```
```  1268 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
```
```  1269   by (fact Sup_inf_eq_bot_iff)
```
```  1270
```
```  1271
```
```  1272 subsection {* Injections and bijections *}
```
```  1273
```
```  1274 lemma inj_on_Inter:
```
```  1275   "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
```
```  1276   unfolding inj_on_def by blast
```
```  1277
```
```  1278 lemma inj_on_INTER:
```
```  1279   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)"
```
```  1280   unfolding inj_on_def by blast
```
```  1281
```
```  1282 lemma inj_on_UNION_chain:
```
```  1283   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
```
```  1284          INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1285   shows "inj_on f (\<Union> i \<in> I. A i)"
```
```  1286 proof -
```
```  1287   {
```
```  1288     fix i j x y
```
```  1289     assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
```
```  1290       and ***: "f x = f y"
```
```  1291     have "x = y"
```
```  1292     proof -
```
```  1293       {
```
```  1294         assume "A i \<le> A j"
```
```  1295         with ** have "x \<in> A j" by auto
```
```  1296         with INJ * ** *** have ?thesis
```
```  1297         by(auto simp add: inj_on_def)
```
```  1298       }
```
```  1299       moreover
```
```  1300       {
```
```  1301         assume "A j \<le> A i"
```
```  1302         with ** have "y \<in> A i" by auto
```
```  1303         with INJ * ** *** have ?thesis
```
```  1304         by(auto simp add: inj_on_def)
```
```  1305       }
```
```  1306       ultimately show ?thesis using CH * by blast
```
```  1307     qed
```
```  1308   }
```
```  1309   then show ?thesis by (unfold inj_on_def UNION_eq) auto
```
```  1310 qed
```
```  1311
```
```  1312 lemma bij_betw_UNION_chain:
```
```  1313   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
```
```  1314          BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
```
```  1315   shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
```
```  1316 proof (unfold bij_betw_def, auto)
```
```  1317   have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
```
```  1318   using BIJ bij_betw_def[of f] by auto
```
```  1319   thus "inj_on f (\<Union> i \<in> I. A i)"
```
```  1320   using CH inj_on_UNION_chain[of I A f] by auto
```
```  1321 next
```
```  1322   fix i x
```
```  1323   assume *: "i \<in> I" "x \<in> A i"
```
```  1324   hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
```
```  1325   thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
```
```  1326 next
```
```  1327   fix i x'
```
```  1328   assume *: "i \<in> I" "x' \<in> A' i"
```
```  1329   hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
```
```  1330   then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
```
```  1331     using * by blast
```
```  1332   then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by blast
```
```  1333 qed
```
```  1334
```
```  1335 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```  1336 lemma image_INT:
```
```  1337    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```  1338     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1339 apply (simp add: inj_on_def, blast)
```
```  1340 done
```
```  1341
```
```  1342 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```  1343   it doesn't matter whether A is empty*)
```
```  1344 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```  1345 apply (simp add: bij_def)
```
```  1346 apply (simp add: inj_on_def surj_def, blast)
```
```  1347 done
```
```  1348
```
```  1349 lemma UNION_fun_upd:
```
```  1350   "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
```
```  1351 by (auto split: if_splits)
```
```  1352
```
```  1353
```
```  1354 subsubsection {* Complement *}
```
```  1355
```
```  1356 lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
```
```  1357   by (fact uminus_INF)
```
```  1358
```
```  1359 lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
```
```  1360   by (fact uminus_SUP)
```
```  1361
```
```  1362
```
```  1363 subsubsection {* Miniscoping and maxiscoping *}
```
```  1364
```
```  1365 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
```
```  1366            and Intersections. *}
```
```  1367
```
```  1368 lemma UN_simps [simp]:
```
```  1369   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
```
```  1370   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
```
```  1371   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
```
```  1372   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
```
```  1373   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
```
```  1374   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
```
```  1375   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
```
```  1376   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
```
```  1377   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
```
```  1378   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
```
```  1379   by auto
```
```  1380
```
```  1381 lemma INT_simps [simp]:
```
```  1382   "\<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)"
```
```  1383   "\<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))"
```
```  1384   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
```
```  1385   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
```
```  1386   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
```
```  1387   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
```
```  1388   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
```
```  1389   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
```
```  1390   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
```
```  1391   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
```
```  1392   by auto
```
```  1393
```
```  1394 lemma UN_ball_bex_simps [simp]:
```
```  1395   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
```
```  1396   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
```
```  1397   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
```
```  1398   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
```
```  1399   by auto
```
```  1400
```
```  1401
```
```  1402 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
```
```  1403
```
```  1404 lemma UN_extend_simps:
```
```  1405   "\<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)))"
```
```  1406   "\<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))"
```
```  1407   "\<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))"
```
```  1408   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
```
```  1409   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
```
```  1410   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
```
```  1411   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
```
```  1412   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
```
```  1413   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
```
```  1414   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
```
```  1415   by auto
```
```  1416
```
```  1417 lemma INT_extend_simps:
```
```  1418   "\<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))"
```
```  1419   "\<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))"
```
```  1420   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
```
```  1421   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
```
```  1422   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
```
```  1423   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
```
```  1424   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
```
```  1425   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
```
```  1426   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
```
```  1427   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
```
```  1428   by auto
```
```  1429
```
```  1430 text {* Finally *}
```
```  1431
```
```  1432 no_notation
```
```  1433   less_eq (infix "\<sqsubseteq>" 50) and
```
```  1434   less (infix "\<sqsubset>" 50)
```
```  1435
```
```  1436 lemmas mem_simps =
```
```  1437   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
```
```  1438   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
```
```  1439   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
```
```  1440
```
```  1441 end
```
```  1442
```