diff --git a/src/HOL/Complete_Lattices.thy b/src/HOL/Complete_Lattices.thy
--- a/src/HOL/Complete_Lattices.thy
+++ b/src/HOL/Complete_Lattices.thy
@@ -1,1364 +1,1364 @@
 (*  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>")
 
 class Sup =
   fixes Sup :: "'a set \<Rightarrow> 'a"  ("\<Squnion>")
 
 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)"
   by (simp add: image_comp)
 
 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)"
   by (simp add: image_def)
 
 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: 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: 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 antisym Inf_greatest Inf_lower)
 
-lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> \<Sqinter>(f ` A)"
-  by (simp cong del: INF_cong_simp)
+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 antisym Sup_least Sup_upper)
 
-lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> \<Squnion>(f ` A)"
-  by (simp cong del: SUP_cong_simp)
+lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> \<Squnion>(f ` A)"
+  by simp
 
-lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
-  by (simp cong del: INF_cong_simp)
+lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"
+  by simp
 
-lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
-  by (simp cong del: SUP_cong_simp)
+lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
+  by simp
 
 lemma Inf_UNIV [simp]: "\<Sqinter>UNIV = \<bottom>"
   by (auto intro!: antisym Inf_lower)
 
 lemma Sup_UNIV [simp]: "\<Squnion>UNIV = \<top>"
   by (auto intro!: antisym Sup_upper)
 
 lemma Inf_eq_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
   by (auto intro: 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: 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 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 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 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!: 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 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!: 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 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 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_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
   by (auto intro: antisym INF_lower INF_greatest)
 
 lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
   by (auto intro: antisym SUP_upper SUP_least)
 
 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 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 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 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 INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
   by simp
 
 lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
   by simp
 
 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"
   by (simp add: UNIV_bool inf_commute)
 
 lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False"
   by (simp add: UNIV_bool sup_commute)
 
 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 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 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 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 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 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)"
   by (simp add: uminus_Inf image_image)
 
 lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
 proof -
   have "\<Squnion>A = - \<Sqinter>(uminus ` A)"
     by (simp add: image_image uminus_INF)
   then show ?thesis by simp
 qed
 
 lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
   by (simp add: uminus_Sup image_image)
 
 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: antisym simp add: min_def fun_eq_iff)
 
 lemma complete_linorder_sup_max: "sup = max"
   by (auto intro: 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"
       by (simp add: less_Sup_iff)
   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"
   by (simp add: fun_eq_iff)
 
 lemma SUP_bool_eq [simp]: "(\<lambda>A f. \<Squnion>(f ` A)) = Bex"
   by (simp add: fun_eq_iff)
 
 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)"
   by (simp add: Inf_fun_def)
 
 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)"
   by (simp add: Sup_fun_def)
 
 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)"
   by (simp add: image_comp)
 
 lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
   by (simp add: image_comp)
 
 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}"
     by (simp add: Inf_set_def image_def)
 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"
   by (simp add: Inter_eq)
 
 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}"
     by (simp add: Sup_set_def image_def)
 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)"
   by (simp add: bind_def UNION_eq)
 
 lemma member_bind [simp]: "x \<in> Set.bind A f \<longleftrightarrow> x \<in> \<Union>(f ` A)"
   by (simp add: bind_UNION)
 
 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 *)
   by (simp add: INT_Int_distrib)
 
 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>
   by (simp add: UN_Un_distrib)
 
 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
       by (auto simp add: inj_on_def)
   next
     assume "A j \<le> A i"
     with ** have "y \<in> A i" by auto
     with inj * ** *** show ?thesis
       by (auto simp add: inj_on_def)
   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 {})"
   by (auto simp add: set_eq_iff)
 
 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
   \<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close>
 
 end
diff --git a/src/HOL/Conditionally_Complete_Lattices.thy b/src/HOL/Conditionally_Complete_Lattices.thy
--- a/src/HOL/Conditionally_Complete_Lattices.thy
+++ b/src/HOL/Conditionally_Complete_Lattices.thy
@@ -1,739 +1,739 @@
 (*  Title:      HOL/Conditionally_Complete_Lattices.thy
     Author:     Amine Chaieb and L C Paulson, University of Cambridge
     Author:     Johannes Hölzl, TU München
     Author:     Luke S. Serafin, Carnegie Mellon University
 *)
 
 section \<open>Conditionally-complete Lattices\<close>
 
 theory Conditionally_Complete_Lattices
 imports Finite_Set Lattices_Big Set_Interval
 begin
 
 context preorder
 begin
 
 definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
 definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
 
 lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
   by (auto simp: bdd_above_def)
 
 lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
   by (auto simp: bdd_below_def)
 
 lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)"
   by force
 
 lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)"
   by force
 
 lemma bdd_above_empty [simp, intro]: "bdd_above {}"
   unfolding bdd_above_def by auto
 
 lemma bdd_below_empty [simp, intro]: "bdd_below {}"
   unfolding bdd_below_def by auto
 
 lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
   by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
 
 lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
   by (metis bdd_below_def order_class.le_neq_trans psubsetD)
 
 lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
   using bdd_above_mono by auto
 
 lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
   using bdd_above_mono by auto
 
 lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
   using bdd_below_mono by auto
 
 lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
   using bdd_below_mono by auto
 
 lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
 
 lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
 
 lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
 
 lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
 
 lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
 
 lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
 
 lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
 
 lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
 
 lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
 
 lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
 
 lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
 
 lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
 
 end
 
 lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A"
   by (rule bdd_aboveI[of _ top]) simp
 
 lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A"
   by (rule bdd_belowI[of _ bot]) simp
 
 lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)"
   by (auto simp: bdd_above_def mono_def)
 
 lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)"
   by (auto simp: bdd_below_def mono_def)
 
 lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)"
   by (auto simp: bdd_above_def bdd_below_def antimono_def)
 
 lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)"
   by (auto simp: bdd_above_def bdd_below_def antimono_def)
 
 lemma
   fixes X :: "'a::ordered_ab_group_add set"
   shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
     and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
   using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
   using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
   by (auto simp: antimono_def image_image)
 
 context lattice
 begin
 
 lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
   by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
 
 lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
   by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
 
 lemma bdd_finite [simp]:
   assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
   using assms by (induct rule: finite_induct, auto)
 
 lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
 proof
   assume "bdd_above (A \<union> B)"
   thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
 next
   assume "bdd_above A \<and> bdd_above B"
   then obtain a b where "\<forall>x\<in>A. x \<le> a" "\<forall>x\<in>B. x \<le> b" unfolding bdd_above_def by auto
   hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
   thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
 qed
 
 lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
 proof
   assume "bdd_below (A \<union> B)"
   thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
 next
   assume "bdd_below A \<and> bdd_below B"
   then obtain a b where "\<forall>x\<in>A. a \<le> x" "\<forall>x\<in>B. b \<le> x" unfolding bdd_below_def by auto
   hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
   thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
 qed
 
 lemma bdd_above_image_sup[simp]:
   "bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)"
 by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
 
 lemma bdd_below_image_inf[simp]:
   "bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)"
 by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
 
 lemma bdd_below_UN[simp]: "finite I \<Longrightarrow> bdd_below (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_below (A i))"
 by (induction I rule: finite.induct) auto
 
 lemma bdd_above_UN[simp]: "finite I \<Longrightarrow> bdd_above (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_above (A i))"
 by (induction I rule: finite.induct) auto
 
 end
 
 
 text \<open>
 
 To avoid name classes with the \<^class>\<open>complete_lattice\<close>-class we prefix \<^const>\<open>Sup\<close> and
 \<^const>\<open>Inf\<close> in theorem names with c.
 
 \<close>
 
 class conditionally_complete_lattice = lattice + Sup + Inf +
   assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
     and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
   assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
     and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
 begin
 
 lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
   by (metis cSup_upper order_trans)
 
 lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
   by (metis cInf_lower order_trans)
 
 lemma cSup_mono: "B \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b \<le> a) \<Longrightarrow> Sup B \<le> Sup A"
   by (metis cSup_least cSup_upper2)
 
 lemma cInf_mono: "B \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b) \<Longrightarrow> Inf A \<le> Inf B"
   by (metis cInf_greatest cInf_lower2)
 
 lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
   by (metis cSup_least cSup_upper subsetD)
 
 lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
   by (metis cInf_greatest cInf_lower subsetD)
 
 lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
   by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
 
 lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
   by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
 
 lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
   by (metis order_trans cSup_upper cSup_least)
 
 lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
   by (metis order_trans cInf_lower cInf_greatest)
 
 lemma cSup_eq_non_empty:
   assumes 1: "X \<noteq> {}"
   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
   shows "Sup X = a"
   by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
 
 lemma cInf_eq_non_empty:
   assumes 1: "X \<noteq> {}"
   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
   shows "Inf X = a"
   by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
 
 lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
   by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
 
 lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
   by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
 
 lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
   by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
 
 lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
   by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
 
 lemma cSup_singleton [simp]: "Sup {x} = x"
   by (intro cSup_eq_maximum) auto
 
 lemma cInf_singleton [simp]: "Inf {x} = x"
   by (intro cInf_eq_minimum) auto
 
 lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
   using cSup_insert[of X] by simp
 
 lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
   using cInf_insert[of X] by simp
 
 lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
 proof (induct X arbitrary: x rule: finite_induct)
   case (insert x X y) then show ?case
     by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
 qed simp
 
 lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
 proof (induct X arbitrary: x rule: finite_induct)
   case (insert x X y) then show ?case
     by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
 qed simp
 
 lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
   by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
 
 lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
   by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
 
 lemma cSup_atMost[simp]: "Sup {..x} = x"
   by (auto intro!: cSup_eq_maximum)
 
 lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
   by (auto intro!: cSup_eq_maximum)
 
 lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
   by (auto intro!: cSup_eq_maximum)
 
 lemma cInf_atLeast[simp]: "Inf {x..} = x"
   by (auto intro!: cInf_eq_minimum)
 
 lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
   by (auto intro!: cInf_eq_minimum)
 
 lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
   by (auto intro!: cInf_eq_minimum)
 
 lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> \<Sqinter>(f ` A) \<le> f x"
   using cInf_lower [of _ "f ` A"] by simp
 
 lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> \<Sqinter>(f ` A)"
   using cInf_greatest [of "f ` A"] by auto
 
 lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> \<Squnion>(f ` A)"
   using cSup_upper [of _ "f ` A"] by simp
 
 lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> \<Squnion>(f ` A) \<le> M"
   using cSup_least [of "f ` A"] by auto
 
 lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> \<Sqinter>(f ` A) \<le> u"
   by (auto intro: cINF_lower order_trans)
 
 lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> \<Squnion>(f ` A)"
   by (auto intro: cSUP_upper order_trans)
 
 lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>x\<in>A. c) = c"
   by (intro antisym cSUP_least) (auto intro: cSUP_upper)
 
 lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>x\<in>A. c) = c"
   by (intro antisym cINF_greatest) (auto intro: cINF_lower)
 
 lemma le_cINF_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> u \<le> \<Sqinter>(f ` A) \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)"
   by (metis cINF_greatest cINF_lower order_trans)
 
 lemma cSUP_le_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> \<Squnion>(f ` A) \<le> u \<longleftrightarrow> (\<forall>x\<in>A. f x \<le> u)"
   by (metis cSUP_least cSUP_upper order_trans)
 
 lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (\<Sqinter>i\<in>A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
   by (metis cINF_lower less_le_trans)
 
 lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (\<Squnion>i\<in>A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
   by (metis cSUP_upper le_less_trans)
 
 lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> \<Sqinter>(f ` insert a A) = inf (f a) (\<Sqinter>(f ` A))"
-  by (simp add: cInf_insert cong del: INF_cong)
+  by (simp add: cInf_insert)
 
 lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> \<Squnion>(f ` insert a A) = sup (f a) (\<Squnion>(f ` A))"
-  by (simp add: cSup_insert cong del: SUP_cong)
+  by (simp add: cSup_insert)
 
 lemma cINF_mono: "B \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> (\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> \<Sqinter>(f ` A) \<le> \<Sqinter>(g ` B)"
   using cInf_mono [of "g ` B" "f ` A"] by auto
 
 lemma cSUP_mono: "A \<noteq> {} \<Longrightarrow> bdd_above (g ` B) \<Longrightarrow> (\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> \<Squnion>(f ` A) \<le> \<Squnion>(g ` B)"
   using cSup_mono [of "f ` A" "g ` B"] by auto
 
 lemma cINF_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below (g ` B) \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> g x \<le> f x) \<Longrightarrow> \<Sqinter>(g ` B) \<le> \<Sqinter>(f ` A)"
   by (rule cINF_mono) auto
 
 lemma cSUP_subset_mono: 
   "\<lbrakk>A \<noteq> {}; bdd_above (g ` B); A \<subseteq> B; \<And>x. x \<in> A \<Longrightarrow> f x \<le> g x\<rbrakk> \<Longrightarrow> \<Squnion> (f ` A) \<le> \<Squnion> (g ` B)"
   by (rule cSUP_mono) auto
 
 lemma less_eq_cInf_inter: "bdd_below A \<Longrightarrow> bdd_below B \<Longrightarrow> A \<inter> B \<noteq> {} \<Longrightarrow> inf (Inf A) (Inf B) \<le> Inf (A \<inter> B)"
   by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
 
 lemma cSup_inter_less_eq: "bdd_above A \<Longrightarrow> bdd_above B \<Longrightarrow> A \<inter> B \<noteq> {} \<Longrightarrow> Sup (A \<inter> B) \<le> sup (Sup A) (Sup B) "
   by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
 
 lemma cInf_union_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> Inf (A \<union> B) = inf (Inf A) (Inf B)"
   by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
 
 lemma cINF_union: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below (f ` B) \<Longrightarrow> \<Sqinter> (f ` (A \<union> B)) = \<Sqinter> (f ` A) \<sqinter> \<Sqinter> (f ` B)"
-  using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un cong del: INF_cong)
+  using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un)
 
 lemma cSup_union_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> Sup (A \<union> B) = sup (Sup A) (Sup B)"
   by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
 
 lemma cSUP_union: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above (f ` B) \<Longrightarrow> \<Squnion> (f ` (A \<union> B)) = \<Squnion> (f ` A) \<squnion> \<Squnion> (f ` B)"
-  using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un cong del: SUP_cong)
+  using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un)
 
 lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> \<Sqinter> (f ` A) \<sqinter> \<Sqinter> (g ` A) = (\<Sqinter>a\<in>A. inf (f a) (g a))"
   by (intro antisym le_infI cINF_greatest cINF_lower2)
      (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
 
 lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> \<Squnion> (f ` A) \<squnion> \<Squnion> (g ` A) = (\<Squnion>a\<in>A. sup (f a) (g a))"
   by (intro antisym le_supI cSUP_least cSUP_upper2)
      (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
 
 lemma cInf_le_cSup:
   "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
   by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
 
 end
 
 instance complete_lattice \<subseteq> conditionally_complete_lattice
   by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
 
 lemma cSup_eq:
   fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
   shows "Sup X = a"
 proof cases
   assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
 qed (intro cSup_eq_non_empty assms)
 
 lemma cInf_eq:
   fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
   shows "Inf X = a"
 proof cases
   assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
 qed (intro cInf_eq_non_empty assms)
 
 class conditionally_complete_linorder = conditionally_complete_lattice + linorder
 begin
 
 lemma less_cSup_iff:
   "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
   by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
 
 lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
   by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
 
 lemma cINF_less_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
   using cInf_less_iff[of "f`A"] by auto
 
 lemma less_cSUP_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
   using less_cSup_iff[of "f`A"] by auto
 
 lemma less_cSupE:
   assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
   by (metis cSup_least assms not_le that)
 
 lemma less_cSupD:
   "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
   by (metis less_cSup_iff not_le_imp_less bdd_above_def)
 
 lemma cInf_lessD:
   "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
   by (metis cInf_less_iff not_le_imp_less bdd_below_def)
 
 lemma complete_interval:
   assumes "a < b" and "P a" and "\<not> P b"
   shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
              (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
 proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x}"], auto)
   show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
     by (rule cSup_upper, auto simp: bdd_above_def)
        (metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le)
 next
   show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
     apply (rule cSup_least)
     apply auto
     apply (metis less_le_not_le)
     apply (metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le)
     done
 next
   fix x
   assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
   show "P x"
     apply (rule less_cSupE [OF lt], auto)
     apply (metis less_le_not_le)
     apply (metis x)
     done
 next
   fix d
     assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
     thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
       by (rule_tac cSup_upper, auto simp: bdd_above_def)
          (metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le)
 qed
 
 end
 
 instance complete_linorder < conditionally_complete_linorder
   ..
 
 lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
   using cSup_eq_Sup_fin[of X] by (simp add: Sup_fin_Max)
 
 lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
   using cInf_eq_Inf_fin[of X] by (simp add: Inf_fin_Min)
 
 lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
   by (auto intro!: cSup_eq_non_empty intro: dense_le)
 
 lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
 
 lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
 
 lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
   by (auto intro!: cInf_eq_non_empty intro: dense_ge)
 
 lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
 
 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
 
 lemma Inf_insert_finite:
   fixes S :: "'a::conditionally_complete_linorder set"
   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
   by (simp add: cInf_eq_Min)
 
 lemma Sup_insert_finite:
   fixes S :: "'a::conditionally_complete_linorder set"
   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
   by (simp add: cSup_insert sup_max)
 
 lemma finite_imp_less_Inf:
   fixes a :: "'a::conditionally_complete_linorder"
   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a < x\<rbrakk> \<Longrightarrow> a < Inf X"
   by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)
 
 lemma finite_less_Inf_iff:
   fixes a :: "'a :: conditionally_complete_linorder"
   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a < Inf X \<longleftrightarrow> (\<forall>x \<in> X. a < x)"
   by (auto simp: cInf_eq_Min)
 
 lemma finite_imp_Sup_less:
   fixes a :: "'a::conditionally_complete_linorder"
   shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a > x\<rbrakk> \<Longrightarrow> a > Sup X"
   by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)
 
 lemma finite_Sup_less_iff:
   fixes a :: "'a :: conditionally_complete_linorder"
   shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a > Sup X \<longleftrightarrow> (\<forall>x \<in> X. a > x)"
   by (auto simp: cSup_eq_Max)
 
 class linear_continuum = conditionally_complete_linorder + dense_linorder +
   assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
 begin
 
 lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
   by (metis UNIV_not_singleton neq_iff)
 
 end
 
 instantiation nat :: conditionally_complete_linorder
 begin
 
 definition "Sup (X::nat set) = Max X"
 definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
 
 lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
 proof
   assume "bdd_above X"
   then obtain z where "X \<subseteq> {.. z}"
     by (auto simp: bdd_above_def)
   then show "finite X"
     by (rule finite_subset) simp
 qed simp
 
 instance
 proof
   fix x :: nat
   fix X :: "nat set"
   show "Inf X \<le> x" if "x \<in> X" "bdd_below X"
     using that by (simp add: Inf_nat_def Least_le)
   show "x \<le> Inf X" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y"
     using that unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex)
   show "x \<le> Sup X" if "x \<in> X" "bdd_above X"
     using that by (simp add: Sup_nat_def bdd_above_nat)
   show "Sup X \<le> x" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x"
   proof -
     from that have "bdd_above X"
       by (auto simp: bdd_above_def)
     with that show ?thesis 
       by (simp add: Sup_nat_def bdd_above_nat)
   qed
 qed
 
 end
 
 lemma Inf_nat_def1:
   fixes K::"nat set"
   assumes "K \<noteq> {}"
   shows "Inf K \<in> K"
 by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI)
 
 
 instantiation int :: conditionally_complete_linorder
 begin
 
 definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
 definition "Inf (X::int set) = - (Sup (uminus ` X))"
 
 instance
 proof
   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
     then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
       by (auto simp: bdd_above_def)
     then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
       by (auto simp: subset_eq)
     have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
     proof
       { fix z assume "z \<in> X"
         have "z \<le> Max (X \<inter> {x..y})"
         proof cases
           assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis
             by (auto intro!: Max_ge)
         next
           assume "\<not> x \<le> z"
           then have "z < x" by simp
           also have "x \<le> Max (X \<inter> {x..y})"
             using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto
           finally show ?thesis by simp
         qed }
       note le = this
       with Max_in[OF *] show ex: "Max (X \<inter> {x..y}) \<in> X \<and> (\<forall>z\<in>X. z \<le> Max (X \<inter> {x..y}))" by auto
 
       fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
       with le have "z \<le> Max (X \<inter> {x..y})"
         by auto
       moreover have "Max (X \<inter> {x..y}) \<le> z"
         using * ex by auto
       ultimately show "z = Max (X \<inter> {x..y})"
         by auto
     qed
     then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
       unfolding Sup_int_def by (rule theI') }
   note Sup_int = this
 
   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
       using Sup_int[of X] by auto }
   note le_Sup = this
   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x" then show "Sup X \<le> x"
       using Sup_int[of X] by (auto simp: bdd_above_def) }
   note Sup_le = this
 
   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
       using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" then show "x \<le> Inf X"
       using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
 qed
 end
 
 lemma interval_cases:
   fixes S :: "'a :: conditionally_complete_linorder set"
   assumes ivl: "\<And>a b x. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> x \<in> S"
   shows "\<exists>a b. S = {} \<or>
     S = UNIV \<or>
     S = {..<b} \<or>
     S = {..b} \<or>
     S = {a<..} \<or>
     S = {a..} \<or>
     S = {a<..<b} \<or>
     S = {a<..b} \<or>
     S = {a..<b} \<or>
     S = {a..b}"
 proof -
   define lower upper where "lower = {x. \<exists>s\<in>S. s \<le> x}" and "upper = {x. \<exists>s\<in>S. x \<le> s}"
   with ivl have "S = lower \<inter> upper"
     by auto
   moreover
   have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
   proof cases
     assume *: "bdd_above S \<and> S \<noteq> {}"
     from * have "upper \<subseteq> {.. Sup S}"
       by (auto simp: upper_def intro: cSup_upper2)
     moreover from * have "{..< Sup S} \<subseteq> upper"
       by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
     ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
       unfolding ivl_disj_un(2)[symmetric] by auto
     then show ?thesis by auto
   next
     assume "\<not> (bdd_above S \<and> S \<noteq> {})"
     then have "upper = UNIV \<or> upper = {}"
       by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
     then show ?thesis
       by auto
   qed
   moreover
   have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
   proof cases
     assume *: "bdd_below S \<and> S \<noteq> {}"
     from * have "lower \<subseteq> {Inf S ..}"
       by (auto simp: lower_def intro: cInf_lower2)
     moreover from * have "{Inf S <..} \<subseteq> lower"
       by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
     ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
       unfolding ivl_disj_un(1)[symmetric] by auto
     then show ?thesis by auto
   next
     assume "\<not> (bdd_below S \<and> S \<noteq> {})"
     then have "lower = UNIV \<or> lower = {}"
       by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
     then show ?thesis
       by auto
   qed
   ultimately show ?thesis
     unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
     by (metis inf_bot_left inf_bot_right inf_top.left_neutral inf_top.right_neutral)
 qed
 
 lemma cSUP_eq_cINF_D:
   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
   assumes eq: "(\<Squnion>x\<in>A. f x) = (\<Sqinter>x\<in>A. f x)"
      and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
      and a: "a \<in> A"
   shows "f a = (\<Sqinter>x\<in>A. f x)"
 apply (rule antisym)
 using a bdd
 apply (auto simp: cINF_lower)
 apply (metis eq cSUP_upper)
 done
 
 lemma cSUP_UNION:
   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
       and bdd_UN: "bdd_above (\<Union>x\<in>A. f ` B x)"
   shows "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) = (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)"
 proof -
   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_above (f ` B x)"
     using bdd_UN by (meson UN_upper bdd_above_mono)
   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<le> M"
     using bdd_UN by (auto simp: bdd_above_def)
   then have bdd2: "bdd_above ((\<lambda>x. \<Squnion>z\<in>B x. f z) ` A)"
     unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2))
   have "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)"
     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd)
   moreover have "(\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z) \<le> (\<Squnion> z \<in> \<Union>x\<in>A. B x. f z)"
     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN)
   ultimately show ?thesis
     by (rule order_antisym)
 qed
 
 lemma cINF_UNION:
   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
       and bdd_UN: "bdd_below (\<Union>x\<in>A. f ` B x)"
   shows "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) = (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)"
 proof -
   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_below (f ` B x)"
     using bdd_UN by (meson UN_upper bdd_below_mono)
   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<ge> M"
     using bdd_UN by (auto simp: bdd_below_def)
   then have bdd2: "bdd_below ((\<lambda>x. \<Sqinter>z\<in>B x. f z) ` A)"
     unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2))
   have "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)"
     using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd)
   moreover have "(\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z) \<le> (\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z)"
     using assms  by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2  simp: bdd bdd_UN bdd2)
   ultimately show ?thesis
     by (rule order_antisym)
 qed
 
 lemma cSup_abs_le:
   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
   shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
   apply (auto simp add: abs_le_iff intro: cSup_least)
   by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)
 
 end