src/HOL/Complete_Lattices.thy
 author desharna Sat, 27 Nov 2021 22:20:27 +0100 changeset 74866 a8927420a48b parent 74337 9c1ad2f04660 permissions -rw-r--r--
merged
```
(*  Title:      HOL/Complete_Lattices.thy
Author:     Tobias Nipkow
Author:     Lawrence C Paulson
Author:     Markus Wenzel
Author:     Florian Haftmann
Author:     Viorel Preoteasa (Complete Distributive Lattices)
*)

section \<open>Complete lattices\<close>

theory Complete_Lattices
imports Fun
begin

subsection \<open>Syntactic infimum and supremum operations\<close>

class Inf =
fixes Inf :: "'a set \<Rightarrow> 'a"  ("\<Sqinter> _" [900] 900)

class Sup =
fixes Sup :: "'a set \<Rightarrow> 'a"  ("\<Squnion> _" [900] 900)

syntax
"_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
"_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _\<in>_./ _)" [0, 0, 10] 10)
"_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
"_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _\<in>_./ _)" [0, 0, 10] 10)

syntax
"_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
"_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
"_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
"_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

translations
"\<Sqinter>x y. f"   \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. f"
"\<Sqinter>x. f"     \<rightleftharpoons> "\<Sqinter>(CONST range (\<lambda>x. f))"
"\<Sqinter>x\<in>A. f"   \<rightleftharpoons> "CONST Inf ((\<lambda>x. f) ` A)"
"\<Squnion>x y. f"   \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. f"
"\<Squnion>x. f"     \<rightleftharpoons> "\<Squnion>(CONST range (\<lambda>x. f))"
"\<Squnion>x\<in>A. f"   \<rightleftharpoons> "CONST Sup ((\<lambda>x. f) `  A)"

context Inf
begin

lemma INF_image: "\<Sqinter> (g ` f ` A) = \<Sqinter> ((g \<circ> f) ` A)"

lemma INF_identity_eq [simp]: "(\<Sqinter>x\<in>A. x) = \<Sqinter>A"
by simp

lemma INF_id_eq [simp]: "\<Sqinter>(id ` A) = \<Sqinter>A"
by simp

lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> \<Sqinter>(C ` A) = \<Sqinter>(D ` B)"

lemma INF_cong_simp:
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> \<Sqinter>(C ` A) = \<Sqinter>(D ` B)"
unfolding simp_implies_def by (fact INF_cong)

end

context Sup
begin

lemma SUP_image: "\<Squnion> (g ` f ` A) = \<Squnion> ((g \<circ> f) ` A)"
by(fact Inf.INF_image)

lemma SUP_identity_eq [simp]: "(\<Squnion>x\<in>A. x) = \<Squnion>A"
by(fact Inf.INF_identity_eq)

lemma SUP_id_eq [simp]: "\<Squnion>(id ` A) = \<Squnion>A"
by(fact Inf.INF_id_eq)

lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> \<Squnion>(C ` A) = \<Squnion>(D ` B)"
by (fact Inf.INF_cong)

lemma SUP_cong_simp:
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> \<Squnion>(C ` A) = \<Squnion>(D ` B)"
by (fact Inf.INF_cong_simp)

end

subsection \<open>Abstract complete lattices\<close>

text \<open>A complete lattice always has a bottom and a top,
so we include them into the following type class,
along with assumptions that define bottom and top
in terms of infimum and supremum.\<close>

class complete_lattice = lattice + Inf + Sup + bot + top +
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<le> x"
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> \<Sqinter>A"
and Sup_upper: "x \<in> A \<Longrightarrow> x \<le> \<Squnion>A"
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> \<Squnion>A \<le> z"
and Inf_empty [simp]: "\<Sqinter>{} = \<top>"
and Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
begin

subclass bounded_lattice
proof
fix a
show "\<bottom> \<le> a"
by (auto intro: Sup_least simp only: Sup_empty [symmetric])
show "a \<le> \<top>"
by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
qed

lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>"
by (auto intro!: class.complete_lattice.intro dual_lattice)
(unfold_locales, (fact Inf_empty Sup_empty Sup_upper Sup_least Inf_lower Inf_greatest)+)

end

context complete_lattice
begin

lemma Sup_eqI:
"(\<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"
by (blast intro: order.antisym Sup_least Sup_upper)

lemma Inf_eqI:
"(\<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"
by (blast intro: order.antisym Inf_greatest Inf_lower)

lemma SUP_eqI:
"(\<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"
using Sup_eqI [of "f ` A" x] by auto

lemma INF_eqI:
"(\<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"
using Inf_eqI [of "f ` A" x] by auto

lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> f i"
using Inf_lower [of _ "f ` A"] by simp

lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<le> f i) \<Longrightarrow> u \<le> (\<Sqinter>i\<in>A. f i)"
using Inf_greatest [of "f ` A"] by auto

lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<le> (\<Squnion>i\<in>A. f i)"
using Sup_upper [of _ "f ` A"] by simp

lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<le> u"
using Sup_least [of "f ` A"] by auto

lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<le> v \<Longrightarrow> \<Sqinter>A \<le> v"
using Inf_lower [of u A] by auto

lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<le> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> u"
using INF_lower [of i A f] by auto

lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<le> u \<Longrightarrow> v \<le> \<Squnion>A"
using Sup_upper [of u A] by auto

lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<le> f i \<Longrightarrow> u \<le> (\<Squnion>i\<in>A. f i)"
using SUP_upper [of i A f] by auto

lemma le_Inf_iff: "b \<le> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<le> a)"
by (auto intro: Inf_greatest dest: Inf_lower)

lemma le_INF_iff: "u \<le> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<le> f i)"
using le_Inf_iff [of _ "f ` A"] by simp

lemma Sup_le_iff: "\<Squnion>A \<le> b \<longleftrightarrow> (\<forall>a\<in>A. a \<le> b)"
by (auto intro: Sup_least dest: Sup_upper)

lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<le> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<le> u)"
using Sup_le_iff [of "f ` A"] by simp

lemma Inf_insert [simp]: "\<Sqinter>(insert a A) = a \<sqinter> \<Sqinter>A"
by (auto intro: le_infI le_infI1 le_infI2 order.antisym Inf_greatest Inf_lower)

lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> \<Sqinter>(f ` A)"
by simp

lemma Sup_insert [simp]: "\<Squnion>(insert a A) = a \<squnion> \<Squnion>A"
by (auto intro: le_supI le_supI1 le_supI2 order.antisym Sup_least Sup_upper)

lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> \<Squnion>(f ` A)"
by simp

lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"
by simp

lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
by simp

lemma Inf_UNIV [simp]: "\<Sqinter>UNIV = \<bottom>"
by (auto intro!: order.antisym Inf_lower)

lemma Sup_UNIV [simp]: "\<Squnion>UNIV = \<top>"
by (auto intro!: order.antisym Sup_upper)

lemma Inf_eq_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
by (auto intro: order.antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Sup_eq_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
by (auto intro: order.antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<le> \<Sqinter>B"
by (auto intro: Inf_greatest Inf_lower)

lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<le> \<Squnion>B"
by (auto intro: Sup_least Sup_upper)

lemma Inf_mono:
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
shows "\<Sqinter>A \<le> \<Sqinter>B"
proof (rule Inf_greatest)
fix b assume "b \<in> B"
with assms obtain a where "a \<in> A" and "a \<le> b" by blast
from \<open>a \<in> A\<close> have "\<Sqinter>A \<le> a" by (rule Inf_lower)
with \<open>a \<le> b\<close> show "\<Sqinter>A \<le> b" by auto
qed

lemma INF_mono: "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<le> (\<Sqinter>n\<in>B. g n)"
using Inf_mono [of "g ` B" "f ` A"] by auto

lemma INF_mono': "(\<And>x. f x \<le> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<le> (\<Sqinter>x\<in>A. g x)"
by (rule INF_mono) auto

lemma Sup_mono:
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
shows "\<Squnion>A \<le> \<Squnion>B"
proof (rule Sup_least)
fix a assume "a \<in> A"
with assms obtain b where "b \<in> B" and "a \<le> b" by blast
from \<open>b \<in> B\<close> have "b \<le> \<Squnion>B" by (rule Sup_upper)
with \<open>a \<le> b\<close> show "a \<le> \<Squnion>B" by auto
qed

lemma SUP_mono: "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<le> (\<Squnion>n\<in>B. g n)"
using Sup_mono [of "f ` A" "g ` B"] by auto

lemma SUP_mono': "(\<And>x. f x \<le> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<le> (\<Squnion>x\<in>A. g x)"
by (rule SUP_mono) auto

lemma INF_superset_mono: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<le> (\<Sqinter>x\<in>B. g x)"
\<comment> \<open>The last inclusion is POSITIVE!\<close>
by (blast intro: INF_mono dest: subsetD)

lemma SUP_subset_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<le> (\<Squnion>x\<in>B. g x)"
by (blast intro: SUP_mono dest: subsetD)

lemma Inf_less_eq:
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<le> u"
and "A \<noteq> {}"
shows "\<Sqinter>A \<le> u"
proof -
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
moreover from \<open>v \<in> A\<close> assms(1) have "v \<le> u" by blast
ultimately show ?thesis by (rule Inf_lower2)
qed

lemma less_eq_Sup:
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<le> v"
and "A \<noteq> {}"
shows "u \<le> \<Squnion>A"
proof -
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast
moreover from \<open>v \<in> A\<close> assms(1) have "u \<le> v" by blast
ultimately show ?thesis by (rule Sup_upper2)
qed

lemma INF_eq:
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
shows "\<Sqinter>(f ` A) = \<Sqinter>(g ` B)"
by (intro order.antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+

lemma SUP_eq:
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
shows "\<Squnion>(f ` A) = \<Squnion>(g ` B)"
by (intro order.antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+

lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<le> \<Sqinter>(A \<inter> B)"
by (auto intro: Inf_greatest Inf_lower)

lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<le> \<Squnion>A \<sqinter> \<Squnion>B "
by (auto intro: Sup_least Sup_upper)

lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
by (rule order.antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)

lemma INF_union: "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
by (auto intro!: order.antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)

lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
by (rule order.antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)

lemma SUP_union: "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
by (auto intro!: order.antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)

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)"
by (rule order.antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)

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")
proof (rule order.antisym)
show "?L \<le> ?R"
by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
show "?R \<le> ?L"
by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
qed

lemma Inf_top_conv [simp]:
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
proof -
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
proof
assume "\<forall>x\<in>A. x = \<top>"
then have "A = {} \<or> A = {\<top>}" by auto
then show "\<Sqinter>A = \<top>" by auto
next
assume "\<Sqinter>A = \<top>"
show "\<forall>x\<in>A. x = \<top>"
proof (rule ccontr)
assume "\<not> (\<forall>x\<in>A. x = \<top>)"
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
then obtain B where "A = insert x B" by blast
with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp
qed
qed
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
qed

lemma INF_top_conv [simp]:
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
using Inf_top_conv [of "B ` A"] by simp_all

lemma Sup_bot_conv [simp]:
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)"
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)"
using dual_complete_lattice
by (rule complete_lattice.Inf_top_conv)+

lemma SUP_bot_conv [simp]:
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
using Sup_bot_conv [of "B ` A"] by simp_all

lemma INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
by (auto intro: order.antisym INF_lower INF_greatest)

lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
by (auto intro: order.antisym SUP_upper SUP_least)

lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"

lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"

lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
by (cases "A = {}") simp_all

lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
by (cases "A = {}") simp_all

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)"
by (iprover intro: INF_lower INF_greatest order_trans order.antisym)

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)"
by (iprover intro: SUP_upper SUP_least order_trans order.antisym)

lemma INF_absorb:
assumes "k \<in> I"
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
proof -
from assms obtain J where "I = insert k J" by blast
then show ?thesis by simp
qed

lemma SUP_absorb:
assumes "k \<in> I"
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
proof -
from assms obtain J where "I = insert k J" by blast
then show ?thesis by simp
qed

lemma INF_inf_const1: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. inf x (f i)) = inf x (\<Sqinter>i\<in>I. f i)"
by (intro order.antisym INF_greatest inf_mono order_refl INF_lower)
(auto intro: INF_lower2 le_infI2 intro!: INF_mono)

lemma INF_inf_const2: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. inf (f i) x) = inf (\<Sqinter>i\<in>I. f i) x"
using INF_inf_const1[of I x f] by (simp add: inf_commute)

lemma less_INF_D:
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A"
shows "y < f i"
proof -
note \<open>y < (\<Sqinter>i\<in>A. f i)\<close>
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close>
by (rule INF_lower)
finally show "y < f i" .
qed

lemma SUP_lessD:
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A"
shows "f i < y"
proof -
have "f i \<le> (\<Squnion>i\<in>A. f i)"
using \<open>i \<in> A\<close> by (rule SUP_upper)
also note \<open>(\<Squnion>i\<in>A. f i) < y\<close>
finally show "f i < y" .
qed

lemma INF_UNIV_bool_expand: "(\<Sqinter>b. A b) = A True \<sqinter> A False"

lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False"

lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
by (blast intro: Sup_upper2 Inf_lower ex_in_conv)

lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> \<Sqinter>(f ` A) \<le> \<Squnion>(f ` A)"
using Inf_le_Sup [of "f ` A"] by simp

lemma INF_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> \<Sqinter>(f ` I) = x"
by (auto intro: INF_eqI)

lemma SUP_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> \<Squnion>(f ` I) = x"
by (auto intro: SUP_eqI)

lemma INF_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> \<Sqinter>(f ` I) = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
by (auto intro: INF_eq_const INF_lower order.antisym)

lemma SUP_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> \<Squnion>(f ` I) = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)"
by (auto intro: SUP_eq_const SUP_upper order.antisym)

end

context complete_lattice
begin
lemma Sup_Inf_le: "Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)}) \<le> Inf (Sup ` A)"
by (rule SUP_least, clarify, rule INF_greatest, simp add: INF_lower2 Sup_upper)
end

class complete_distrib_lattice = complete_lattice +
assumes Inf_Sup_le: "Inf (Sup ` A) \<le> Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)})"
begin

lemma Inf_Sup: "Inf (Sup ` A) = Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)})"
by (rule order.antisym, rule Inf_Sup_le, rule Sup_Inf_le)

subclass distrib_lattice
proof
fix a b c
show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)"
proof (rule order.antisym, simp_all, safe)
show "b \<sqinter> c \<le> a \<squnion> b"
by (rule le_infI1, simp)
show "b \<sqinter> c \<le> a \<squnion> c"
by (rule le_infI2, simp)
have [simp]: "a \<sqinter> c \<le> a \<squnion> b \<sqinter> c"
by (rule le_infI1, simp)
have [simp]: "b \<sqinter> a \<le> a \<squnion> b \<sqinter> c"
by (rule le_infI2, simp)
have "\<Sqinter>(Sup ` {{a, b}, {a, c}}) =
\<Squnion>(Inf ` {f ` {{a, b}, {a, c}} | f. \<forall>Y\<in>{{a, b}, {a, c}}. f Y \<in> Y})"
by (rule Inf_Sup)
from this show "(a \<squnion> b) \<sqinter> (a \<squnion> c) \<le> a \<squnion> b \<sqinter> c"
apply simp
by (rule SUP_least, safe, simp_all)
qed
qed
end

context complete_lattice
begin
context
fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
assumes "mono f"
begin

lemma mono_Inf: "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)"
using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)

lemma mono_Sup: "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)"
using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)

lemma mono_INF: "f (\<Sqinter>i\<in>I. A i) \<le> (\<Sqinter>x\<in>I. f (A x))"
by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower)

lemma mono_SUP: "(\<Squnion>x\<in>I. f (A x)) \<le> f (\<Squnion>i\<in>I. A i)"
by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper)

end

end

class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
begin

lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
proof (rule order.antisym)
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
qed

lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"

lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
proof -
have "\<Squnion>A = - \<Sqinter>(uminus ` A)"
then show ?thesis by simp
qed

lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"

end

class complete_linorder = linorder + complete_lattice
begin

lemma dual_complete_linorder:
"class.complete_linorder Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>"
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)

lemma complete_linorder_inf_min: "inf = min"
by (auto intro: order.antisym simp add: min_def fun_eq_iff)

lemma complete_linorder_sup_max: "sup = max"
by (auto intro: order.antisym simp add: max_def fun_eq_iff)

lemma Inf_less_iff: "\<Sqinter>S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
by (simp add: not_le [symmetric] le_Inf_iff)

lemma INF_less_iff: "(\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
by (simp add: Inf_less_iff [of "f ` A"])

lemma less_Sup_iff: "a < \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
by (simp add: not_le [symmetric] Sup_le_iff)

lemma less_SUP_iff: "a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
by (simp add: less_Sup_iff [of _ "f ` A"])

lemma Sup_eq_top_iff [simp]: "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
proof
assume *: "\<Squnion>A = \<top>"
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
unfolding * [symmetric]
proof (intro allI impI)
fix x
assume "x < \<Squnion>A"
then show "\<exists>i\<in>A. x < i"
qed
next
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
show "\<Squnion>A = \<top>"
proof (rule ccontr)
assume "\<Squnion>A \<noteq> \<top>"
with top_greatest [of "\<Squnion>A"] have "\<Squnion>A < \<top>"
unfolding le_less by auto
with * have "\<Squnion>A < \<Squnion>A"
unfolding less_Sup_iff by auto
then show False by auto
qed
qed

lemma SUP_eq_top_iff [simp]: "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
using Sup_eq_top_iff [of "f ` A"] by simp

lemma Inf_eq_bot_iff [simp]: "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
using dual_complete_linorder
by (rule complete_linorder.Sup_eq_top_iff)

lemma INF_eq_bot_iff [simp]: "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
using Inf_eq_bot_iff [of "f ` A"] by simp

lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
proof safe
fix y
assume "x \<ge> \<Sqinter>A" "y > x"
then have "y > \<Sqinter>A" by auto
then show "\<exists>a\<in>A. y > a"
unfolding Inf_less_iff .
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)

lemma INF_le_iff: "\<Sqinter>(f ` A) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
using Inf_le_iff [of "f ` A"] by simp

lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
proof safe
fix y
assume "x \<le> \<Squnion>A" "y < x"
then have "y < \<Squnion>A" by auto
then show "\<exists>a\<in>A. y < a"
unfolding less_Sup_iff .
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)

lemma le_SUP_iff: "x \<le> \<Squnion>(f ` A) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
using le_Sup_iff [of _ "f ` A"] by simp

end

subsection \<open>Complete lattice on \<^typ>\<open>bool\<close>\<close>

instantiation bool :: complete_lattice
begin

definition [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"

definition [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"

instance
by standard (auto intro: bool_induct)

end

lemma not_False_in_image_Ball [simp]: "False \<notin> P ` A \<longleftrightarrow> Ball A P"
by auto

lemma True_in_image_Bex [simp]: "True \<in> P ` A \<longleftrightarrow> Bex A P"
by auto

lemma INF_bool_eq [simp]: "(\<lambda>A f. \<Sqinter>(f ` A)) = Ball"

lemma SUP_bool_eq [simp]: "(\<lambda>A f. \<Squnion>(f ` A)) = Bex"

instance bool :: complete_boolean_algebra
by (standard, fastforce)

subsection \<open>Complete lattice on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>

instantiation "fun" :: (type, Inf) Inf
begin

definition "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"

lemma Inf_apply [simp, code]: "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"

instance ..

end

instantiation "fun" :: (type, Sup) Sup
begin

definition "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"

lemma Sup_apply [simp, code]: "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"

instance ..

end

instantiation "fun" :: (type, complete_lattice) complete_lattice
begin

instance
by standard (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)

end

lemma INF_apply [simp]: "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"

lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"

subsection \<open>Complete lattice on unary and binary predicates\<close>

lemma Inf1_I: "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a"
by auto

lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
by simp

lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
by simp

lemma Inf2_I: "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b"
by auto

lemma Inf1_D: "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a"
by auto

lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
by simp

lemma Inf2_D: "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b"
by auto

lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
by simp

lemma Inf1_E:
assumes "(\<Sqinter>A) a"
obtains "P a" | "P \<notin> A"
using assms by auto

lemma INF1_E:
assumes "(\<Sqinter>x\<in>A. B x) b"
obtains "B a b" | "a \<notin> A"
using assms by auto

lemma Inf2_E:
assumes "(\<Sqinter>A) a b"
obtains "r a b" | "r \<notin> A"
using assms by auto

lemma INF2_E:
assumes "(\<Sqinter>x\<in>A. B x) b c"
obtains "B a b c" | "a \<notin> A"
using assms by auto

lemma Sup1_I: "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a"
by auto

lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
by auto

lemma Sup2_I: "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b"
by auto

lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
by auto

lemma Sup1_E:
assumes "(\<Squnion>A) a"
obtains P where "P \<in> A" and "P a"
using assms by auto

lemma SUP1_E:
assumes "(\<Squnion>x\<in>A. B x) b"
obtains x where "x \<in> A" and "B x b"
using assms by auto

lemma Sup2_E:
assumes "(\<Squnion>A) a b"
obtains r where "r \<in> A" "r a b"
using assms by auto

lemma SUP2_E:
assumes "(\<Squnion>x\<in>A. B x) b c"
obtains x where "x \<in> A" "B x b c"
using assms by auto

subsection \<open>Complete lattice on \<^typ>\<open>_ set\<close>\<close>

instantiation "set" :: (type) complete_lattice
begin

definition "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"

definition "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"

instance
by standard (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)

end

subsubsection \<open>Inter\<close>

abbreviation Inter :: "'a set set \<Rightarrow> 'a set"  ("\<Inter>")
where "\<Inter>S \<equiv> \<Sqinter>S"

lemma Inter_eq: "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
proof (rule set_eqI)
fix x
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
qed

lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
by (unfold Inter_eq) blast

lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"

text \<open>
\<^medskip> A ``destruct'' rule -- every \<^term>\<open>X\<close> in \<^term>\<open>C\<close>
contains \<^term>\<open>A\<close> as an element, but \<^prop>\<open>A \<in> X\<close> can hold when
\<^prop>\<open>X \<in> C\<close> does not!  This rule is analogous to \<open>spec\<close>.
\<close>

lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
by auto

lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
\<comment> \<open>``Classical'' elimination rule -- does not require proving
\<^prop>\<open>X \<in> C\<close>.\<close>
unfolding Inter_eq by blast

lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
by (fact Inf_lower)

lemma Inter_subset: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
by (fact Inf_less_eq)

lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A"
by (fact Inf_greatest)

lemma Inter_empty: "\<Inter>{} = UNIV"
by (fact Inf_empty) (* already simp *)

lemma Inter_UNIV: "\<Inter>UNIV = {}"
by (fact Inf_UNIV) (* already simp *)

lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
by (fact Inf_insert) (* already simp *)

lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
by (fact less_eq_Inf_inter)

lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
by (fact Inf_union_distrib)

lemma Inter_UNIV_conv [simp]:
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
by (fact Inf_top_conv)+

lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
by (fact Inf_superset_mono)

subsubsection \<open>Intersections of families\<close>

syntax (ASCII)
"_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3INT _./ _)" [0, 10] 10)
"_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)

syntax
"_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
"_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)

syntax (latex output)
"_INTER1"     :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set"           ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"_INTER"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set"  ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

translations
"\<Inter>x y. f"  \<rightleftharpoons> "\<Inter>x. \<Inter>y. f"
"\<Inter>x. f"    \<rightleftharpoons> "\<Inter>(CONST range (\<lambda>x. f))"
"\<Inter>x\<in>A. f"  \<rightleftharpoons> "CONST Inter ((\<lambda>x. f) ` A)"

lemma INTER_eq: "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
by (auto intro!: INF_eqI)

lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
using Inter_iff [of _ "B ` A"] by simp

lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
by auto

lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
by auto

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"
\<comment> \<open>"Classical" elimination -- by the Excluded Middle on \<^prop>\<open>a\<in>A\<close>.\<close>
by auto

lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
by blast

lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
by blast

lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
by (fact INF_lower)

lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
by (fact INF_greatest)

lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
by (fact INF_empty)

lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
by (fact INF_absorb)

lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
by (fact le_INF_iff)

lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> \<Inter> (B ` A)"
by (fact INF_insert)

lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
by (fact INF_union)

lemma INT_insert_distrib: "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
by blast

lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
by (fact INF_constant)

lemma INTER_UNIV_conv:
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
by (fact INF_top_conv)+ (* already simp *)

lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
by (fact INF_UNIV_bool_expand)

lemma INT_anti_mono: "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)"
\<comment> \<open>The last inclusion is POSITIVE!\<close>
by (fact INF_superset_mono)

lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
by blast

lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
by blast

subsubsection \<open>Union\<close>

abbreviation Union :: "'a set set \<Rightarrow> 'a set"  ("\<Union>")
where "\<Union>S \<equiv> \<Squnion>S"

lemma Union_eq: "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
proof (rule set_eqI)
fix x
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
qed

lemma Union_iff [simp]: "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
by (unfold Union_eq) blast

lemma UnionI [intro]: "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
\<comment> \<open>The order of the premises presupposes that \<^term>\<open>C\<close> is rigid;
\<^term>\<open>A\<close> may be flexible.\<close>
by auto

lemma UnionE [elim!]: "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
by auto

lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
by (fact Sup_upper)

lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
by (fact Sup_least)

lemma Union_empty: "\<Union>{} = {}"
by (fact Sup_empty) (* already simp *)

lemma Union_UNIV: "\<Union>UNIV = UNIV"
by (fact Sup_UNIV) (* already simp *)

lemma Union_insert: "\<Union>(insert a B) = a \<union> \<Union>B"
by (fact Sup_insert) (* already simp *)

lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
by (fact Sup_union_distrib)

lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
by (fact Sup_inter_less_eq)

lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
by (fact Sup_bot_conv) (* already simp *)

lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
by (fact Sup_bot_conv) (* already simp *)

lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
by blast

lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
by blast

lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
by (fact Sup_subset_mono)

lemma Union_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y) \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
by blast

lemma disjnt_inj_on_iff:
"\<lbrakk>inj_on f (\<Union>\<A>); X \<in> \<A>; Y \<in> \<A>\<rbrakk> \<Longrightarrow> disjnt (f ` X) (f ` Y) \<longleftrightarrow> disjnt X Y"
apply (auto simp: disjnt_def)
using inj_on_eq_iff by fastforce

lemma disjnt_Union1 [simp]: "disjnt (\<Union>\<A>) B \<longleftrightarrow> (\<forall>A \<in> \<A>. disjnt A B)"
by (auto simp: disjnt_def)

lemma disjnt_Union2 [simp]: "disjnt B (\<Union>\<A>) \<longleftrightarrow> (\<forall>A \<in> \<A>. disjnt B A)"
by (auto simp: disjnt_def)

subsubsection \<open>Unions of families\<close>

syntax (ASCII)
"_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
"_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)

syntax
"_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
"_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)

syntax (latex output)
"_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

translations
"\<Union>x y. f"   \<rightleftharpoons> "\<Union>x. \<Union>y. f"
"\<Union>x. f"     \<rightleftharpoons> "\<Union>(CONST range (\<lambda>x. f))"
"\<Union>x\<in>A. f"   \<rightleftharpoons> "CONST Union ((\<lambda>x. f) ` A)"

text \<open>
Note the difference between ordinary syntax of indexed
unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>)
and their \LaTeX\ rendition: \<^term>\<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>.
\<close>

lemma disjoint_UN_iff: "disjnt A (\<Union>i\<in>I. B i) \<longleftrightarrow> (\<forall>i\<in>I. disjnt A (B i))"
by (auto simp: disjnt_def)

lemma UNION_eq: "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
by (auto intro!: SUP_eqI)

lemma bind_UNION [code]: "Set.bind A f = \<Union>(f ` A)"

lemma member_bind [simp]: "x \<in> Set.bind A f \<longleftrightarrow> x \<in> \<Union>(f ` A)"

lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
by blast

lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
using Union_iff [of _ "B ` A"] by simp

lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
\<comment> \<open>The order of the premises presupposes that \<^term>\<open>A\<close> is rigid;
\<^term>\<open>b\<close> may be flexible.\<close>
by auto

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"
by auto

lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
by (fact SUP_upper)

lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
by (fact SUP_least)

lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
by blast

lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
by blast

lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
by (fact SUP_empty)

lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
by (fact SUP_bot) (* already simp *)

lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
by (fact SUP_absorb)

lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> \<Union>(B ` A)"
by (fact SUP_insert)

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)"
by (fact SUP_union)

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)"
by blast

lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
by (fact SUP_le_iff)

lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
by (fact SUP_constant)

lemma UNION_singleton_eq_range: "(\<Union>x\<in>A. {f x}) = f ` A"
by blast

lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
by blast

lemma UNION_empty_conv:
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
by (fact SUP_bot_conv)+ (* already simp *)

lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
by blast

lemma ball_UN: "(\<forall>z \<in> \<Union>(B ` A). P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
by blast

lemma bex_UN: "(\<exists>z \<in> \<Union>(B ` A). P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
by blast

lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
by safe (auto simp add: if_split_mem2)

lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
by (fact SUP_UNIV_bool_expand)

lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
by blast

lemma UN_mono:
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
by (fact SUP_subset_mono)

lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
by blast

lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
by blast

lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
\<comment> \<open>NOT suitable for rewriting\<close>
by blast

lemma image_UN: "f ` \<Union>(B ` A) = (\<Union>x\<in>A. f ` B x)"
by blast

lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
by blast

lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on ((`) f) A"
unfolding inj_on_def by blast

subsubsection \<open>Distributive laws\<close>

lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
by blast

lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
by blast

lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
by blast

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)"
by (rule sym) (rule INF_inf_distrib)

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)"
by (rule sym) (rule SUP_sup_distrib)

lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"  (* FIXME drop *)

lemma Int_Inter_eq: "A \<inter> \<Inter>\<B> = (if \<B>={} then A else (\<Inter>B\<in>\<B>. A \<inter> B))"
"\<Inter>\<B> \<inter> A = (if \<B>={} then A else (\<Inter>B\<in>\<B>. B \<inter> A))"
by auto

lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"  (* FIXME drop *)
\<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close>
\<comment> \<open>Union of a family of unions\<close>

lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
by blast

lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
\<comment> \<open>Halmos, Naive Set Theory, page 35.\<close>
by blast

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)"
by blast

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)"
by blast

lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
by blast

lemma SUP_UNION: "(\<Squnion>x\<in>(\<Union>y\<in>A. g y). f x) = (\<Squnion>y\<in>A. \<Squnion>x\<in>g y. f x :: _ :: complete_lattice)"
by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+

subsection \<open>Injections and bijections\<close>

lemma inj_on_Inter: "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)"
unfolding inj_on_def by blast

lemma inj_on_INTER: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)"
unfolding inj_on_def by safe simp

lemma inj_on_UNION_chain:
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
and inj: "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)"
shows "inj_on f (\<Union>i \<in> I. A i)"
proof -
have "x = y"
if *: "i \<in> I" "j \<in> I"
and **: "x \<in> A i" "y \<in> A j"
and ***: "f x = f y"
for i j x y
using chain [OF *]
proof
assume "A i \<le> A j"
with ** have "x \<in> A j" by auto
with inj * ** *** show ?thesis
next
assume "A j \<le> A i"
with ** have "y \<in> A i" by auto
with inj * ** *** show ?thesis
qed
then show ?thesis
by (unfold inj_on_def UNION_eq) auto
qed

lemma bij_betw_UNION_chain:
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
and bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)"
unfolding bij_betw_def
proof safe
have "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)"
using bij bij_betw_def[of f] by auto
then show "inj_on f (\<Union>(A ` I))"
using chain inj_on_UNION_chain[of I A f] by auto
next
fix i x
assume *: "i \<in> I" "x \<in> A i"
with bij have "f x \<in> A' i"
by (auto simp: bij_betw_def)
with * show "f x \<in> \<Union>(A' ` I)" by blast
next
fix i x'
assume *: "i \<in> I" "x' \<in> A' i"
with bij have "\<exists>x \<in> A i. x' = f x"
unfolding bij_betw_def by blast
with * have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
by blast
then show "x' \<in> f ` \<Union>(A ` I)"
by blast
qed

(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
lemma image_INT: "inj_on f C \<Longrightarrow> \<forall>x\<in>A. B x \<subseteq> C \<Longrightarrow> j \<in> A \<Longrightarrow> f ` (\<Inter>(B ` A)) = (\<Inter>x\<in>A. f ` B x)"
by (auto simp add: inj_on_def) blast

lemma bij_image_INT: "bij f \<Longrightarrow> f ` (\<Inter>(B ` A)) = (\<Inter>x\<in>A. f ` B x)"
by (auto simp: bij_def inj_def surj_def) blast

lemma UNION_fun_upd: "\<Union>(A(i := B) ` J) = \<Union>(A ` (J - {i})) \<union> (if i \<in> J then B else {})"

lemma bij_betw_Pow:
assumes "bij_betw f A B"
shows "bij_betw (image f) (Pow A) (Pow B)"
proof -
from assms have "inj_on f A"
by (rule bij_betw_imp_inj_on)
then have "inj_on f (\<Union>(Pow A))"
by simp
then have "inj_on (image f) (Pow A)"
by (rule inj_on_image)
then have "bij_betw (image f) (Pow A) (image f ` Pow A)"
by (rule inj_on_imp_bij_betw)
moreover from assms have "f ` A = B"
by (rule bij_betw_imp_surj_on)
then have "image f ` Pow A = Pow B"
by (rule image_Pow_surj)
ultimately show ?thesis by simp
qed

subsubsection \<open>Complement\<close>

lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
by blast

lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
by blast

subsubsection \<open>Miniscoping and maxiscoping\<close>

text \<open>\<^medskip> Miniscoping: pushing in quantifiers and big Unions and Intersections.\<close>

lemma UN_simps [simp]:
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
"\<And>A B C. (\<Union>z\<in>(\<Union>(B ` A)). C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
"\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
by auto

lemma INT_simps [simp]:
"\<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)"
"\<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))"
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
"\<And>A B C. (\<Inter>z\<in>(\<Union>(B ` A)). C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
by auto

lemma UN_ball_bex_simps [simp]:
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
"\<And>A B P. (\<forall>x\<in>(\<Union>(B ` A)). P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
"\<And>A B P. (\<exists>x\<in>(\<Union>(B ` A)). P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
by auto

text \<open>\<^medskip> Maxiscoping: pulling out big Unions and Intersections.\<close>

lemma UN_extend_simps:
"\<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)))"
"\<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))"
"\<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))"
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>(\<Union>(B ` A)). C z)"
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
by auto

lemma INT_extend_simps:
"\<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))"
"\<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))"
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>(\<Union>(B ` A)). C z)"
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
by auto

text \<open>Finally\<close>

lemmas mem_simps =
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff