
haftmann@44103:  (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
wenzelm@11979: 
haftmann@44104: header {* Complete lattices *}
haftmann@32077: 
haftmann@44860: theory Complete_Lattices
haftmann@32139: imports Set
haftmann@32139: begin
haftmann@32077: 
haftmann@32077: notation
haftmann@34007:   less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691:   less (infix "\<sqsubset>" 50)
haftmann@32077: 
haftmann@32139: 
haftmann@32879: subsection {* Syntactic infimum and supremum operations *}
haftmann@32879: 
haftmann@32879: class Inf =
haftmann@32879:   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
haftmann@32879: 
haftmann@32879: class Sup =
haftmann@32879:   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
haftmann@32879: 
haftmann@46691: 
haftmann@32139: subsection {* Abstract complete lattices *}
haftmann@32139: 
haftmann@52729: text {* A complete lattice always has a bottom and a top,
haftmann@52729: so we include them into the following type class,
haftmann@52729: along with assumptions that define bottom and top
haftmann@52729: in terms of infimum and supremum. *}
haftmann@52729: 
haftmann@52729: class complete_lattice = lattice + Inf + Sup + bot + top +
haftmann@32077:   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
haftmann@32077:      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
haftmann@32077:   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
haftmann@32077:      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
haftmann@52729:   assumes Inf_empty [simp]: "\<Sqinter>{} = \<top>"
haftmann@52729:   assumes Sup_empty [simp]: "\<Squnion>{} = \<bottom>"
haftmann@32077: begin
haftmann@32077: 
haftmann@52729: subclass bounded_lattice
haftmann@52729: proof
haftmann@52729:   fix a
haftmann@52729:   show "\<bottom> \<le> a" by (auto intro: Sup_least simp only: Sup_empty [symmetric])
haftmann@52729:   show "a \<le> \<top>" by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
haftmann@52729: qed
haftmann@52729: 
haftmann@32678: lemma dual_complete_lattice:
krauss@44845:   "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@52729:   by (auto intro!: class.complete_lattice.intro dual_lattice)
haftmann@52729:     (unfold_locales, (fact Inf_empty Sup_empty
haftmann@34007:         Sup_upper Sup_least Inf_lower Inf_greatest)+)
haftmann@32678: 
haftmann@44040: definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@44040:   INF_def: "INFI A f = \<Sqinter>(f ` A)"
haftmann@44040: 
haftmann@44040: definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@44040:   SUP_def: "SUPR A f = \<Squnion>(f ` A)"
haftmann@44040: 
haftmann@44040: text {*
haftmann@44040:   Note: must use names @{const INFI} and @{const SUPR} here instead of
haftmann@44040:   @{text INF} and @{text SUP} to allow the following syntax coexist
haftmann@44040:   with the plain constant names.
haftmann@44040: *}
haftmann@44040: 
haftmann@44040: end
haftmann@44040: 
haftmann@44040: syntax
haftmann@44040:   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@44040:   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
haftmann@44040:   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@44040:   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
haftmann@44040: 
haftmann@44040: syntax (xsymbols)
haftmann@44040:   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@44040:   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@44040:   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@44040:   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@44040: 
haftmann@44040: translations
haftmann@44040:   "INF x y. B"   == "INF x. INF y. B"
haftmann@44040:   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
haftmann@44040:   "INF x. B"     == "INF x:CONST UNIV. B"
haftmann@44040:   "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@44040:   "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@44040:   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
haftmann@44040:   "SUP x. B"     == "SUP x:CONST UNIV. B"
haftmann@44040:   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@44040: 
haftmann@44040: print_translation {*
haftmann@44040:   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
haftmann@44040:     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
haftmann@44040: *} -- {* to avoid eta-contraction of body *}
haftmann@44040: 
haftmann@44040: context complete_lattice
haftmann@44040: begin
haftmann@32077: 
blanchet@54147: lemma INF_foundation_dual:
haftmann@44040:   "complete_lattice.SUPR Inf = INFI"
huffman@44921:   by (simp add: fun_eq_iff INF_def
huffman@44921:     complete_lattice.SUP_def [OF dual_complete_lattice])
haftmann@44040: 
blanchet@54147: lemma SUP_foundation_dual:
haftmann@44040:   "complete_lattice.INFI Sup = SUPR"
huffman@44921:   by (simp add: fun_eq_iff SUP_def
huffman@44921:     complete_lattice.INF_def [OF dual_complete_lattice])
haftmann@44040: 
hoelzl@51328: lemma Sup_eqI:
hoelzl@51328:   "(\<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"
hoelzl@51328:   by (blast intro: antisym Sup_least Sup_upper)
hoelzl@51328: 
hoelzl@51328: lemma Inf_eqI:
hoelzl@51328:   "(\<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"
hoelzl@51328:   by (blast intro: antisym Inf_greatest Inf_lower)
hoelzl@51328: 
hoelzl@51328: lemma SUP_eqI:
hoelzl@51328:   "(\<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"
hoelzl@51328:   unfolding SUP_def by (rule Sup_eqI) auto
hoelzl@51328: 
hoelzl@51328: lemma INF_eqI:
hoelzl@51328:   "(\<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"
hoelzl@51328:   unfolding INF_def by (rule Inf_eqI) auto
hoelzl@51328: 
haftmann@44103: lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
haftmann@44040:   by (auto simp add: INF_def intro: Inf_lower)
haftmann@44040: 
haftmann@44103: lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
haftmann@44103:   by (auto simp add: INF_def intro: Inf_greatest)
haftmann@44103: 
haftmann@44103: lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
haftmann@44040:   by (auto simp add: SUP_def intro: Sup_upper)
haftmann@44040: 
haftmann@44103: lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
haftmann@44040:   by (auto simp add: SUP_def intro: Sup_least)
haftmann@44040: 
haftmann@44040: lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
haftmann@44040:   using Inf_lower [of u A] by auto
haftmann@44040: 
haftmann@44103: lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
haftmann@44103:   using INF_lower [of i A f] by auto
haftmann@44040: 
haftmann@44040: lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
haftmann@44040:   using Sup_upper [of u A] by auto
haftmann@44040: 
haftmann@44103: lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
haftmann@44103:   using SUP_upper [of i A f] by auto
haftmann@44040: 
noschinl@44918: lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
haftmann@44040:   by (auto intro: Inf_greatest dest: Inf_lower)
haftmann@44040: 
noschinl@44918: lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
haftmann@44040:   by (auto simp add: INF_def le_Inf_iff)
haftmann@44040: 
noschinl@44918: lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
haftmann@44040:   by (auto intro: Sup_least dest: Sup_upper)
haftmann@44040: 
noschinl@44918: lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
haftmann@44040:   by (auto simp add: SUP_def Sup_le_iff)
haftmann@32077: 
haftmann@52729: lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
haftmann@52729:   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
haftmann@52729: 
haftmann@52729: lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
haftmann@52729:   by (simp add: INF_def)
haftmann@52729: 
haftmann@52729: lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
haftmann@52729:   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
haftmann@52729: 
haftmann@52729: lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
haftmann@52729:   by (simp add: SUP_def)
haftmann@32077: 
huffman@44067: lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>"
haftmann@44040:   by (simp add: INF_def)
haftmann@44040: 
huffman@44067: lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
haftmann@44040:   by (simp add: SUP_def)
haftmann@44040: 
haftmann@41080: lemma Inf_UNIV [simp]:
haftmann@41080:   "\<Sqinter>UNIV = \<bottom>"
haftmann@44040:   by (auto intro!: antisym Inf_lower)
haftmann@41080: 
haftmann@41080: lemma Sup_UNIV [simp]:
haftmann@41080:   "\<Squnion>UNIV = \<top>"
haftmann@44040:   by (auto intro!: antisym Sup_upper)
haftmann@41080: 
noschinl@44918: lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
huffman@44068:   by (simp add: INF_def image_image)
huffman@44068: 
noschinl@44918: lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
huffman@44068:   by (simp add: SUP_def image_image)
huffman@44068: 
haftmann@44040: lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
haftmann@44040:   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@44040: 
haftmann@44040: lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
haftmann@44040:   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@44040: 
haftmann@43899: lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
haftmann@43899:   by (auto intro: Inf_greatest Inf_lower)
haftmann@43899: 
haftmann@43899: lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
haftmann@43899:   by (auto intro: Sup_least Sup_upper)
haftmann@43899: 
haftmann@44041: lemma INF_cong:
haftmann@44041:   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
haftmann@44041:   by (simp add: INF_def image_def)
haftmann@44041: 
haftmann@44041: lemma SUP_cong:
haftmann@44041:   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
haftmann@44041:   by (simp add: SUP_def image_def)
haftmann@44041: 
hoelzl@38705: lemma Inf_mono:
hoelzl@41971:   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
haftmann@43741:   shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
hoelzl@38705: proof (rule Inf_greatest)
hoelzl@38705:   fix b assume "b \<in> B"
hoelzl@41971:   with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
haftmann@43741:   from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
haftmann@43741:   with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
hoelzl@38705: qed
hoelzl@38705: 
haftmann@44041: lemma INF_mono:
haftmann@44041:   "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
noschinl@44918:   unfolding INF_def by (rule Inf_mono) fast
haftmann@44041: 
haftmann@41082: lemma Sup_mono:
hoelzl@41971:   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
haftmann@43741:   shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
haftmann@41082: proof (rule Sup_least)
haftmann@41082:   fix a assume "a \<in> A"
hoelzl@41971:   with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
haftmann@43741:   from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
haftmann@43741:   with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
haftmann@41082: qed
haftmann@32077: 
haftmann@44041: lemma SUP_mono:
haftmann@44041:   "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
noschinl@44918:   unfolding SUP_def by (rule Sup_mono) fast
haftmann@44041: 
haftmann@44041: lemma INF_superset_mono:
haftmann@44041:   "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
haftmann@44041:   -- {* The last inclusion is POSITIVE! *}
haftmann@44041:   by (blast intro: INF_mono dest: subsetD)
haftmann@44041: 
haftmann@44041: lemma SUP_subset_mono:
haftmann@44041:   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
haftmann@44041:   by (blast intro: SUP_mono dest: subsetD)
haftmann@44041: 
haftmann@43868: lemma Inf_less_eq:
haftmann@43868:   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43868:     and "A \<noteq> {}"
haftmann@43868:   shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43868: proof -
haftmann@43868:   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
wenzelm@53374:   moreover from `v \<in> A` assms(1) have "v \<sqsubseteq> u" by blast
haftmann@43868:   ultimately show ?thesis by (rule Inf_lower2)
haftmann@43868: qed
haftmann@43868: 
haftmann@43868: lemma less_eq_Sup:
haftmann@43868:   assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v"
haftmann@43868:     and "A \<noteq> {}"
haftmann@43868:   shows "u \<sqsubseteq> \<Squnion>A"
haftmann@43868: proof -
haftmann@43868:   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
wenzelm@53374:   moreover from `v \<in> A` assms(1) have "u \<sqsubseteq> v" by blast
haftmann@43868:   ultimately show ?thesis by (rule Sup_upper2)
haftmann@43868: qed
haftmann@43868: 
hoelzl@51328: lemma SUPR_eq:
hoelzl@51328:   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j"
hoelzl@51328:   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i"
hoelzl@51328:   shows "(SUP i:A. f i) = (SUP j:B. g j)"
hoelzl@51328:   by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+
hoelzl@51328: 
hoelzl@51328: lemma INFI_eq:
hoelzl@51328:   assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j"
hoelzl@51328:   assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i"
hoelzl@51328:   shows "(INF i:A. f i) = (INF j:B. g j)"
hoelzl@51328:   by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+
hoelzl@51328: 
haftmann@43899: lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43868:   by (auto intro: Inf_greatest Inf_lower)
haftmann@43868: 
haftmann@43899: lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B "
haftmann@43868:   by (auto intro: Sup_least Sup_upper)
haftmann@43868: 
haftmann@43868: lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43868:   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@43868: 
haftmann@44041: lemma INF_union:
haftmann@44041:   "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@44103:   by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
haftmann@44041: 
haftmann@43868: lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
haftmann@43868:   by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
haftmann@43868: 
haftmann@44041: lemma SUP_union:
haftmann@44041:   "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@44103:   by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
haftmann@44041: 
haftmann@44041: 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)"
haftmann@44103:   by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
haftmann@44041: 
noschinl@44918: 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")
noschinl@44918: proof (rule antisym)
noschinl@44918:   show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
noschinl@44918: next
noschinl@44918:   show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
noschinl@44918: qed
haftmann@44041: 
blanchet@54147: lemma Inf_top_conv [simp]:
haftmann@43868:   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868:   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868: proof -
haftmann@43868:   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868:   proof
haftmann@43868:     assume "\<forall>x\<in>A. x = \<top>"
haftmann@43868:     then have "A = {} \<or> A = {\<top>}" by auto
noschinl@44919:     then show "\<Sqinter>A = \<top>" by auto
haftmann@43868:   next
haftmann@43868:     assume "\<Sqinter>A = \<top>"
haftmann@43868:     show "\<forall>x\<in>A. x = \<top>"
haftmann@43868:     proof (rule ccontr)
haftmann@43868:       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43868:       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43868:       then obtain B where "A = insert x B" by blast
noschinl@44919:       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp
haftmann@43868:     qed
haftmann@43868:   qed
haftmann@43868:   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43868: qed
haftmann@43868: 
noschinl@44918: lemma INF_top_conv [simp]:
haftmann@44041:  "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@44041:  "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
noschinl@44919:   by (auto simp add: INF_def)
haftmann@44041: 
blanchet@54147: lemma Sup_bot_conv [simp]:
haftmann@43868:   "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
haftmann@43868:   "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
huffman@44920:   using dual_complete_lattice
huffman@44920:   by (rule complete_lattice.Inf_top_conv)+
haftmann@43868: 
noschinl@44918: lemma SUP_bot_conv [simp]:
haftmann@44041:  "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@44041:  "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
noschinl@44919:   by (auto simp add: SUP_def)
haftmann@44041: 
haftmann@43865: lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
haftmann@44103:   by (auto intro: antisym INF_lower INF_greatest)
haftmann@32077: 
haftmann@43870: lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
haftmann@44103:   by (auto intro: antisym SUP_upper SUP_least)
haftmann@43870: 
noschinl@44918: lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
huffman@44921:   by (cases "A = {}") simp_all
haftmann@43900: 
noschinl@44918: lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
huffman@44921:   by (cases "A = {}") simp_all
haftmann@43900: 
haftmann@43865: 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)"
haftmann@44103:   by (iprover intro: INF_lower INF_greatest order_trans antisym)
haftmann@43865: 
haftmann@43870: 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)"
haftmann@44103:   by (iprover intro: SUP_upper SUP_least order_trans antisym)
haftmann@43870: 
haftmann@43871: lemma INF_absorb:
haftmann@43868:   assumes "k \<in> I"
haftmann@43868:   shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868: proof -
haftmann@43868:   from assms obtain J where "I = insert k J" by blast
haftmann@43868:   then show ?thesis by (simp add: INF_insert)
haftmann@43868: qed
haftmann@43868: 
haftmann@43871: lemma SUP_absorb:
haftmann@43871:   assumes "k \<in> I"
haftmann@43871:   shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871: proof -
haftmann@43871:   from assms obtain J where "I = insert k J" by blast
haftmann@43871:   then show ?thesis by (simp add: SUP_insert)
haftmann@43871: qed
haftmann@43871: 
haftmann@43871: lemma INF_constant:
haftmann@43868:   "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
huffman@44921:   by simp
haftmann@43868: 
haftmann@43871: lemma SUP_constant:
haftmann@43871:   "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
huffman@44921:   by simp
haftmann@43871: 
haftmann@43943: lemma less_INF_D:
haftmann@43943:   assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943: proof -
haftmann@43943:   note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943:   also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@44103:     by (rule INF_lower)
haftmann@43943:   finally show "y < f i" .
haftmann@43943: qed
haftmann@43943: 
haftmann@43943: lemma SUP_lessD:
haftmann@43943:   assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943: proof -
haftmann@43943:   have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@44103:     by (rule SUP_upper)
haftmann@43943:   also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943:   finally show "f i < y" .
haftmann@43943: qed
haftmann@43943: 
haftmann@43873: lemma INF_UNIV_bool_expand:
haftmann@43868:   "(\<Sqinter>b. A b) = A True \<sqinter> A False"
huffman@44921:   by (simp add: UNIV_bool INF_insert inf_commute)
haftmann@43868: 
haftmann@43873: lemma SUP_UNIV_bool_expand:
haftmann@43871:   "(\<Squnion>b. A b) = A True \<squnion> A False"
huffman@44921:   by (simp add: UNIV_bool SUP_insert sup_commute)
haftmann@43871: 
hoelzl@51328: lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@51328:   by (blast intro: Sup_upper2 Inf_lower ex_in_conv)
hoelzl@51328: 
hoelzl@51328: lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f"
hoelzl@51328:   unfolding INF_def SUP_def by (rule Inf_le_Sup) auto
hoelzl@51328: 
haftmann@32077: end
haftmann@32077: 
haftmann@44024: class complete_distrib_lattice = complete_lattice +
haftmann@44039:   assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024:   assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024: begin
haftmann@44024: 
haftmann@44039: lemma sup_INF:
haftmann@44039:   "a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)"
haftmann@44039:   by (simp add: INF_def sup_Inf image_image)
haftmann@44039: 
haftmann@44039: lemma inf_SUP:
haftmann@44039:   "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
haftmann@44039:   by (simp add: SUP_def inf_Sup image_image)
haftmann@44039: 
haftmann@44032: lemma dual_complete_distrib_lattice:
krauss@44845:   "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@44024:   apply (rule class.complete_distrib_lattice.intro)
haftmann@44024:   apply (fact dual_complete_lattice)
haftmann@44024:   apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032:   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032:   done
haftmann@44024: 
haftmann@44322: subclass distrib_lattice proof
haftmann@44024:   fix a b c
haftmann@44024:   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
noschinl@44919:   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def)
haftmann@44024: qed
haftmann@44024: 
haftmann@44039: lemma Inf_sup:
haftmann@44039:   "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
haftmann@44039:   by (simp add: sup_Inf sup_commute)
haftmann@44039: 
haftmann@44039: lemma Sup_inf:
haftmann@44039:   "\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)"
haftmann@44039:   by (simp add: inf_Sup inf_commute)
haftmann@44039: 
haftmann@44039: lemma INF_sup: 
haftmann@44039:   "(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)"
haftmann@44039:   by (simp add: sup_INF sup_commute)
haftmann@44039: 
haftmann@44039: lemma SUP_inf:
haftmann@44039:   "(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)"
haftmann@44039:   by (simp add: inf_SUP inf_commute)
haftmann@44039: 
haftmann@44039: lemma Inf_sup_eq_top_iff:
haftmann@44039:   "(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)"
haftmann@44039:   by (simp only: Inf_sup INF_top_conv)
haftmann@44039: 
haftmann@44039: lemma Sup_inf_eq_bot_iff:
haftmann@44039:   "(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)"
haftmann@44039:   by (simp only: Sup_inf SUP_bot_conv)
haftmann@44039: 
haftmann@44039: lemma INF_sup_distrib2:
haftmann@44039:   "(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)"
haftmann@44039:   by (subst INF_commute) (simp add: sup_INF INF_sup)
haftmann@44039: 
haftmann@44039: lemma SUP_inf_distrib2:
haftmann@44039:   "(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)"
haftmann@44039:   by (subst SUP_commute) (simp add: inf_SUP SUP_inf)
haftmann@44039: 
haftmann@44024: end
haftmann@44024: 
haftmann@44032: class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873: begin
haftmann@43873: 
haftmann@43943: lemma dual_complete_boolean_algebra:
krauss@44845:   "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032:   by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943: 
haftmann@43873: lemma uminus_Inf:
haftmann@43873:   "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873: proof (rule antisym)
haftmann@43873:   show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873:     by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873:   show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873:     by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873: qed
haftmann@43873: 
haftmann@44041: lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@44041:   by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@44041: 
haftmann@43873: lemma uminus_Sup:
haftmann@43873:   "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873: proof -
haftmann@43873:   have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873:   then show ?thesis by simp
haftmann@43873: qed
haftmann@43873:   
haftmann@43873: lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873:   by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873: 
haftmann@43873: end
haftmann@43873: 
haftmann@43940: class complete_linorder = linorder + complete_lattice
haftmann@43940: begin
haftmann@43940: 
haftmann@43943: lemma dual_complete_linorder:
krauss@44845:   "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
haftmann@43943:   by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943: 
haftmann@51386: lemma complete_linorder_inf_min: "inf = min"
haftmann@51540:   by (auto intro: antisym simp add: min_def fun_eq_iff)
haftmann@51386: 
haftmann@51386: lemma complete_linorder_sup_max: "sup = max"
haftmann@51540:   by (auto intro: antisym simp add: max_def fun_eq_iff)
haftmann@51386: 
noschinl@44918: lemma Inf_less_iff:
haftmann@43940:   "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940:   unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940: 
noschinl@44918: lemma INF_less_iff:
haftmann@44041:   "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@44041:   unfolding INF_def Inf_less_iff by auto
haftmann@44041: 
noschinl@44918: lemma less_Sup_iff:
haftmann@43940:   "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940:   unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940: 
noschinl@44918: lemma less_SUP_iff:
haftmann@43940:   "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43940:   unfolding SUP_def less_Sup_iff by auto
haftmann@43940: 
noschinl@44918: lemma Sup_eq_top_iff [simp]:
haftmann@43943:   "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943: proof
haftmann@43943:   assume *: "\<Squnion>A = \<top>"
haftmann@43943:   show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943:   proof (intro allI impI)
haftmann@43943:     fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943:       unfolding less_Sup_iff by auto
haftmann@43943:   qed
haftmann@43943: next
haftmann@43943:   assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943:   show "\<Squnion>A = \<top>"
haftmann@43943:   proof (rule ccontr)
haftmann@43943:     assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943:     with top_greatest [of "\<Squnion>A"]
haftmann@43943:     have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943:     then have "\<Squnion>A < \<Squnion>A"
haftmann@43943:       using * unfolding less_Sup_iff by auto
haftmann@43943:     then show False by auto
haftmann@43943:   qed
haftmann@43943: qed
haftmann@43943: 
noschinl@44918: lemma SUP_eq_top_iff [simp]:
haftmann@44041:   "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
noschinl@44919:   unfolding SUP_def by auto
haftmann@44041: 
noschinl@44918: lemma Inf_eq_bot_iff [simp]:
haftmann@43943:   "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
huffman@44920:   using dual_complete_linorder
huffman@44920:   by (rule complete_linorder.Sup_eq_top_iff)
haftmann@43943: 
noschinl@44918: lemma INF_eq_bot_iff [simp]:
haftmann@43967:   "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
noschinl@44919:   unfolding INF_def by auto
haftmann@43967: 
hoelzl@51328: lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
hoelzl@51328: proof safe
hoelzl@51328:   fix y assume "x \<le> \<Squnion>A" "y < x"
hoelzl@51328:   then have "y < \<Squnion>A" by auto
hoelzl@51328:   then show "\<exists>a\<in>A. y < a"
hoelzl@51328:     unfolding less_Sup_iff .
hoelzl@51328: qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper)
hoelzl@51328: 
hoelzl@51328: lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
hoelzl@51328:   unfolding le_Sup_iff SUP_def by simp
hoelzl@51328: 
hoelzl@51328: lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
hoelzl@51328: proof safe
hoelzl@51328:   fix y assume "x \<ge> \<Sqinter>A" "y > x"
hoelzl@51328:   then have "y > \<Sqinter>A" by auto
hoelzl@51328:   then show "\<exists>a\<in>A. y > a"
hoelzl@51328:     unfolding Inf_less_iff .
hoelzl@51328: qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower)
hoelzl@51328: 
hoelzl@51328: lemma INF_le_iff:
hoelzl@51328:   "INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
hoelzl@51328:   unfolding Inf_le_iff INF_def by simp
hoelzl@51328: 
haftmann@51386: subclass complete_distrib_lattice
haftmann@51386: proof
haftmann@51386:   fix a and B
haftmann@51386:   show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@51386:     by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper)
haftmann@51386:       (auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff
haftmann@51386:         le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min)
haftmann@51386: qed
haftmann@51386: 
haftmann@43940: end
haftmann@43940: 
hoelzl@51341: 
haftmann@46631: subsection {* Complete lattice on @{typ bool} *}
haftmann@32077: 
haftmann@44024: instantiation bool :: complete_lattice
haftmann@32077: begin
haftmann@32077: 
haftmann@32077: definition
haftmann@46154:   [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
haftmann@32077: 
haftmann@32077: definition
haftmann@46154:   [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
haftmann@32077: 
haftmann@32077: instance proof
haftmann@44322: qed (auto intro: bool_induct)
haftmann@32077: 
haftmann@32077: end
haftmann@32077: 
haftmann@49905: lemma not_False_in_image_Ball [simp]:
haftmann@49905:   "False \<notin> P ` A \<longleftrightarrow> Ball A P"
haftmann@49905:   by auto
haftmann@49905: 
haftmann@49905: lemma True_in_image_Bex [simp]:
haftmann@49905:   "True \<in> P ` A \<longleftrightarrow> Bex A P"
haftmann@49905:   by auto
haftmann@49905: 
haftmann@43873: lemma INF_bool_eq [simp]:
haftmann@32120:   "INFI = Ball"
haftmann@49905:   by (simp add: fun_eq_iff INF_def)
haftmann@32120: 
haftmann@43873: lemma SUP_bool_eq [simp]:
haftmann@32120:   "SUPR = Bex"
haftmann@49905:   by (simp add: fun_eq_iff SUP_def)
haftmann@32120: 
haftmann@44032: instance bool :: complete_boolean_algebra proof
haftmann@44322: qed (auto intro: bool_induct)
haftmann@44024: 
haftmann@46631: 
haftmann@46631: subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *}
haftmann@46631: 
haftmann@32077: instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077: begin
haftmann@32077: 
haftmann@32077: definition
haftmann@44024:   "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080: 
noschinl@46882: lemma Inf_apply [simp, code]:
haftmann@44024:   "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080:   by (simp add: Inf_fun_def)
haftmann@32077: 
haftmann@32077: definition
haftmann@44024:   "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080: 
noschinl@46882: lemma Sup_apply [simp, code]:
haftmann@44024:   "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080:   by (simp add: Sup_fun_def)
haftmann@32077: 
haftmann@32077: instance proof
noschinl@46884: qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)
haftmann@32077: 
haftmann@32077: end
haftmann@32077: 
noschinl@46882: lemma INF_apply [simp]:
haftmann@41080:   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
noschinl@46884:   by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def)
hoelzl@38705: 
noschinl@46882: lemma SUP_apply [simp]:
haftmann@41080:   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
noschinl@46884:   by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def)
haftmann@32077: 
haftmann@44024: instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
noschinl@46884: qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image)
haftmann@44024: 
haftmann@43873: instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873: 
haftmann@46631: 
haftmann@46631: subsection {* Complete lattice on unary and binary predicates *}
haftmann@46631: 
haftmann@46631: lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
noschinl@46884:   by simp
haftmann@46631: 
haftmann@46631: lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
noschinl@46884:   by simp
haftmann@46631: 
haftmann@46631: lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
noschinl@46884:   by simp
haftmann@46631: 
haftmann@46631: lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
noschinl@46884:   by simp
haftmann@46631: 
haftmann@46631: lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
noschinl@46884:   by auto
haftmann@46631: 
haftmann@46631: 
haftmann@46631: subsection {* Complete lattice on @{typ "_ set"} *}
haftmann@46631: 
haftmann@45960: instantiation "set" :: (type) complete_lattice
haftmann@45960: begin
haftmann@45960: 
haftmann@45960: definition
haftmann@45960:   "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960: 
haftmann@45960: definition
haftmann@45960:   "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}"
haftmann@45960: 
haftmann@45960: instance proof
haftmann@51386: qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)
haftmann@45960: 
haftmann@45960: end
haftmann@45960: 
haftmann@45960: instance "set" :: (type) complete_boolean_algebra
haftmann@45960: proof
haftmann@45960: qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def)
haftmann@45960:   
haftmann@32077: 
haftmann@46631: subsubsection {* Inter *}
haftmann@41082: 
haftmann@41082: abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082:   "Inter S \<equiv> \<Sqinter>S"
haftmann@41082:   
haftmann@41082: notation (xsymbols)
haftmann@52141:   Inter  ("\<Inter>_" [900] 900)
haftmann@41082: 
haftmann@41082: lemma Inter_eq:
haftmann@41082:   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082: proof (rule set_eqI)
haftmann@41082:   fix x
haftmann@41082:   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082:     by auto
haftmann@41082:   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@45960:     by (simp add: Inf_set_def image_def)
haftmann@41082: qed
haftmann@41082: 
blanchet@54147: lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082:   by (unfold Inter_eq) blast
haftmann@41082: 
haftmann@43741: lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082:   by (simp add: Inter_eq)
haftmann@41082: 
haftmann@41082: text {*
haftmann@41082:   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741:   contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741:   @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082: *}
haftmann@41082: 
haftmann@43741: lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082:   by auto
haftmann@41082: 
haftmann@43741: lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082:   -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741:     @{prop "X \<in> C"}. *}
haftmann@41082:   by (unfold Inter_eq) blast
haftmann@41082: 
haftmann@43741: lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740:   by (fact Inf_lower)
haftmann@43740: 
haftmann@41082: lemma Inter_subset:
haftmann@43755:   "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740:   by (fact Inf_less_eq)
haftmann@41082: 
haftmann@43755: lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740:   by (fact Inf_greatest)
haftmann@41082: 
huffman@44067: lemma Inter_empty: "\<Inter>{} = UNIV"
huffman@44067:   by (fact Inf_empty) (* already simp *)
haftmann@41082: 
huffman@44067: lemma Inter_UNIV: "\<Inter>UNIV = {}"
huffman@44067:   by (fact Inf_UNIV) (* already simp *)
haftmann@41082: 
huffman@44920: lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
huffman@44920:   by (fact Inf_insert) (* already simp *)
haftmann@41082: 
haftmann@41082: lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899:   by (fact less_eq_Inf_inter)
haftmann@41082: 
haftmann@41082: lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756:   by (fact Inf_union_distrib)
haftmann@43756: 
blanchet@54147: lemma Inter_UNIV_conv [simp]:
haftmann@43741:   "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741:   "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801:   by (fact Inf_top_conv)+
haftmann@41082: 
haftmann@43741: lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899:   by (fact Inf_superset_mono)
haftmann@41082: 
haftmann@41082: 
haftmann@46631: subsubsection {* Intersections of families *}
haftmann@41082: 
haftmann@41082: abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082:   "INTER \<equiv> INFI"
haftmann@41082: 
haftmann@43872: text {*
haftmann@43872:   Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872:   to allow the following syntax coexist with the plain constant name.
haftmann@43872: *}
haftmann@43872: 
haftmann@41082: syntax
haftmann@41082:   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082:   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082: 
haftmann@41082: syntax (xsymbols)
haftmann@41082:   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082:   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082: 
haftmann@41082: syntax (latex output)
haftmann@41082:   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082:   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082: 
haftmann@41082: translations
haftmann@41082:   "INT x y. B"  == "INT x. INT y. B"
haftmann@41082:   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082:   "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082:   "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082: 
haftmann@41082: print_translation {*
wenzelm@42284:   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082: *} -- {* to avoid eta-contraction of body *}
haftmann@41082: 
haftmann@44085: lemma INTER_eq:
haftmann@41082:   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@44085:   by (auto simp add: INF_def)
haftmann@41082: 
haftmann@41082: lemma Inter_image_eq [simp]:
haftmann@41082:   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872:   by (rule sym) (fact INF_def)
haftmann@41082: 
haftmann@43817: lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@44085:   by (auto simp add: INF_def image_def)
haftmann@41082: 
haftmann@43817: lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@44085:   by (auto simp add: INF_def image_def)
haftmann@41082: 
haftmann@43852: lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
haftmann@41082:   by auto
haftmann@41082: 
haftmann@43852: 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"
haftmann@43852:   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@44085:   by (auto simp add: INF_def image_def)
haftmann@41082: 
haftmann@41082: lemma INT_cong [cong]:
haftmann@43854:   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
haftmann@43865:   by (fact INF_cong)
haftmann@41082: 
haftmann@41082: lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@43817: lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@44103:   by (fact INF_lower)
haftmann@41082: 
haftmann@43817: lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@44103:   by (fact INF_greatest)
haftmann@41082: 
huffman@44067: lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@44085:   by (fact INF_empty)
haftmann@43854: 
haftmann@43817: lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872:   by (fact INF_absorb)
haftmann@41082: 
haftmann@43854: lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082:   by (fact le_INF_iff)
haftmann@41082: 
haftmann@41082: lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865:   by (fact INF_insert)
haftmann@43865: 
haftmann@43865: lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@43865:   by (fact INF_union)
haftmann@43865: 
haftmann@43865: lemma INT_insert_distrib:
haftmann@43865:   "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865:   by blast
haftmann@43854: 
haftmann@41082: lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865:   by (fact INF_constant)
haftmann@43865: 
huffman@44920: lemma INTER_UNIV_conv:
haftmann@43817:  "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817:  "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
huffman@44920:   by (fact INF_top_conv)+ (* already simp *)
haftmann@43865: 
haftmann@43865: lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873:   by (fact INF_UNIV_bool_expand)
haftmann@43865: 
haftmann@43865: lemma INT_anti_mono:
haftmann@43900:   "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)"
haftmann@43865:   -- {* The last inclusion is POSITIVE! *}
haftmann@43940:   by (fact INF_superset_mono)
haftmann@41082: 
haftmann@41082: lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@43817: lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: 
haftmann@46631: subsubsection {* Union *}
haftmann@32115: 
haftmann@32587: abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587:   "Union S \<equiv> \<Squnion>S"
haftmann@32115: 
haftmann@32115: notation (xsymbols)
haftmann@52141:   Union  ("\<Union>_" [900] 900)
haftmann@32115: 
haftmann@32135: lemma Union_eq:
haftmann@32135:   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302: proof (rule set_eqI)
haftmann@32115:   fix x
haftmann@32135:   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115:     by auto
haftmann@32135:   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@45960:     by (simp add: Sup_set_def image_def)
haftmann@32115: qed
haftmann@32115: 
blanchet@54147: lemma Union_iff [simp]:
haftmann@32115:   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115:   by (unfold Union_eq) blast
haftmann@32115: 
haftmann@32115: lemma UnionI [intro]:
haftmann@32115:   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115:   -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115:     @{term A} may be flexible. *}
haftmann@32115:   by auto
haftmann@32115: 
haftmann@32115: lemma UnionE [elim!]:
haftmann@43817:   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115:   by auto
haftmann@32115: 
haftmann@43817: lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901:   by (fact Sup_upper)
haftmann@32135: 
haftmann@43817: lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901:   by (fact Sup_least)
haftmann@32135: 
huffman@44920: lemma Union_empty: "\<Union>{} = {}"
huffman@44920:   by (fact Sup_empty) (* already simp *)
haftmann@32135: 
huffman@44920: lemma Union_UNIV: "\<Union>UNIV = UNIV"
huffman@44920:   by (fact Sup_UNIV) (* already simp *)
haftmann@32135: 
huffman@44920: lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B"
huffman@44920:   by (fact Sup_insert) (* already simp *)
haftmann@32135: 
haftmann@43817: lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901:   by (fact Sup_union_distrib)
haftmann@32135: 
haftmann@32135: lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901:   by (fact Sup_inter_less_eq)
haftmann@32135: 
blanchet@54147: lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920:   by (fact Sup_bot_conv) (* already simp *)
haftmann@32135: 
blanchet@54147: lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
huffman@44920:   by (fact Sup_bot_conv) (* already simp *)
haftmann@32135: 
haftmann@32135: lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43817: lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901:   by (fact Sup_subset_mono)
haftmann@32135: 
haftmann@32115: 
haftmann@46631: subsubsection {* Unions of families *}
haftmann@32077: 
haftmann@32606: abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606:   "UNION \<equiv> SUPR"
haftmann@32077: 
haftmann@43872: text {*
haftmann@43872:   Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872:   to allow the following syntax coexist with the plain constant name.
haftmann@43872: *}
haftmann@43872: 
haftmann@32077: syntax
wenzelm@35115:   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364:   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077: 
haftmann@32077: syntax (xsymbols)
wenzelm@35115:   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364:   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077: 
haftmann@32077: syntax (latex output)
wenzelm@35115:   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364:   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077: 
haftmann@32077: translations
haftmann@32077:   "UN x y. B"   == "UN x. UN y. B"
haftmann@32077:   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077:   "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077:   "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077: 
haftmann@32077: text {*
haftmann@32077:   Note the difference between ordinary xsymbol syntax of indexed
wenzelm@53015:   unions and intersections (e.g.\ @{text"\<Union>a\<^sub>1\<in>A\<^sub>1. B"})
wenzelm@53015:   and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. The
haftmann@32077:   former does not make the index expression a subscript of the
haftmann@32077:   union/intersection symbol because this leads to problems with nested
haftmann@32077:   subscripts in Proof General.
haftmann@32077: *}
haftmann@32077: 
wenzelm@35115: print_translation {*
wenzelm@42284:   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115: *} -- {* to avoid eta-contraction of body *}
haftmann@32077: 
blanchet@54147: lemma UNION_eq:
haftmann@32135:   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@44085:   by (auto simp add: SUP_def)
huffman@44920: 
haftmann@45960: lemma bind_UNION [code]:
haftmann@45960:   "Set.bind A f = UNION A f"
haftmann@45960:   by (simp add: bind_def UNION_eq)
haftmann@45960: 
haftmann@46036: lemma member_bind [simp]:
haftmann@46036:   "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f "
haftmann@46036:   by (simp add: bind_UNION)
haftmann@46036: 
haftmann@32115: lemma Union_image_eq [simp]:
haftmann@43817:   "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
huffman@44920:   by (rule sym) (fact SUP_def)
huffman@44920: 
haftmann@46036: lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)"
haftmann@44085:   by (auto simp add: SUP_def image_def)
wenzelm@11979: 
haftmann@43852: lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979:   -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979:     @{term b} may be flexible. *}
wenzelm@11979:   by auto
wenzelm@11979: 
haftmann@43852: 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"
haftmann@44085:   by (auto simp add: SUP_def image_def)
clasohm@923: 
wenzelm@11979: lemma UN_cong [cong]:
haftmann@43900:   "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900:   by (fact SUP_cong)
wenzelm@11979: 
berghofe@29691: lemma strong_UN_cong:
haftmann@43900:   "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900:   by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691: 
haftmann@43817: lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077:   by blast
haftmann@32077: 
haftmann@43817: lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@44103:   by (fact SUP_upper)
haftmann@32135: 
haftmann@43817: lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@44103:   by (fact SUP_least)
haftmann@32135: 
blanchet@54147: lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43817: lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135:   by blast
haftmann@32135: 
blanchet@54147: lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}"
haftmann@44085:   by (fact SUP_empty)
haftmann@32135: 
huffman@44920: lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}"
huffman@44920:   by (fact SUP_bot) (* already simp *)
haftmann@32135: 
haftmann@43817: lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900:   by (fact SUP_absorb)
haftmann@32135: 
haftmann@32135: lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900:   by (fact SUP_insert)
haftmann@32135: 
haftmann@44085: 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)"
haftmann@43900:   by (fact SUP_union)
haftmann@32135: 
haftmann@43967: 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)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629:   by (fact SUP_le_iff)
haftmann@32135: 
haftmann@32135: lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900:   by (fact SUP_constant)
haftmann@32135: 
haftmann@43944: lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135:   by blast
haftmann@32135: 
huffman@44920: lemma UNION_empty_conv:
haftmann@43817:   "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817:   "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
huffman@44920:   by (fact SUP_bot_conv)+ (* already simp *)
haftmann@32135: 
blanchet@54147: lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43900: lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43900: lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135:   by (auto simp add: split_if_mem2)
haftmann@32135: 
haftmann@43817: lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900:   by (fact SUP_UNIV_bool_expand)
haftmann@32135: 
haftmann@32135: lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_mono:
haftmann@43817:   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135:     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940:   by (fact SUP_subset_mono)
haftmann@32135: 
haftmann@43817: lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43817: lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43817: lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135:   -- {* NOT suitable for rewriting *}
haftmann@32135:   by blast
haftmann@32135: 
haftmann@43817: lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817:   by blast
haftmann@32135: 
haftmann@45013: lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@45013:   by blast
haftmann@45013: 
wenzelm@11979: 
haftmann@46631: subsubsection {* Distributive laws *}
wenzelm@12897: 
wenzelm@12897: lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032:   by (fact inf_Sup)
wenzelm@12897: 
haftmann@44039: lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44039:   by (fact sup_Inf)
haftmann@44039: 
wenzelm@12897: lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
haftmann@44039:   by (fact Sup_inf)
haftmann@44039: 
haftmann@44039: 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)"
haftmann@44039:   by (rule sym) (rule INF_inf_distrib)
haftmann@44039: 
haftmann@44039: 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)"
haftmann@44039:   by (rule sym) (rule SUP_sup_distrib)
haftmann@44039: 
haftmann@44039: lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
haftmann@44039:   by (simp only: INT_Int_distrib INF_def)
wenzelm@12897: 
haftmann@43817: lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897:   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897:   -- {* Union of a family of unions *}
haftmann@44039:   by (simp only: UN_Un_distrib SUP_def)
wenzelm@12897: 
haftmann@44039: lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
haftmann@44039:   by (fact sup_INF)
wenzelm@12897: 
wenzelm@12897: lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897:   -- {* Halmos, Naive Set Theory, page 35. *}
haftmann@44039:   by (fact inf_SUP)
wenzelm@12897: 
wenzelm@12897: 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)"
haftmann@44039:   by (fact SUP_inf_distrib2)
wenzelm@12897: 
wenzelm@12897: 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)"
haftmann@44039:   by (fact INF_sup_distrib2)
haftmann@44039: 
haftmann@44039: lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@44039:   by (fact Sup_inf_eq_bot_iff)
wenzelm@12897: 
wenzelm@12897: 
haftmann@46631: subsubsection {* Complement *}
haftmann@32135: 
haftmann@43873: lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873:   by (fact uminus_INF)
wenzelm@12897: 
haftmann@43873: lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873:   by (fact uminus_SUP)
wenzelm@12897: 
wenzelm@12897: 
haftmann@46631: subsubsection {* Miniscoping and maxiscoping *}
wenzelm@12897: 
paulson@13860: text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860:            and Intersections. *}
wenzelm@12897: 
wenzelm@12897: lemma UN_simps [simp]:
haftmann@43817:   "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@44032:   "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
haftmann@43852:   "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
haftmann@44032:   "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852:   "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852:   "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852:   "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852:   "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852:   "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
haftmann@43831:   "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897:   by auto
wenzelm@12897: 
wenzelm@12897: lemma INT_simps [simp]:
haftmann@44032:   "\<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)"
haftmann@43831:   "\<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))"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
haftmann@43817:   "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852:   "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852:   "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
haftmann@43852:   "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897:   by auto
wenzelm@12897: 
blanchet@54147: lemma UN_ball_bex_simps [simp]:
haftmann@43852:   "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967:   "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
haftmann@43852:   "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852:   "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
wenzelm@12897:   by auto
wenzelm@12897: 
haftmann@43943: 
paulson@13860: text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860: 
paulson@13860: lemma UN_extend_simps:
haftmann@43817:   "\<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)))"
haftmann@44032:   "\<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))"
haftmann@43852:   "\<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))"
haftmann@43852:   "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852:   "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817:   "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817:   "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852:   "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852:   "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
haftmann@43831:   "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860:   by auto
paulson@13860: 
paulson@13860: lemma INT_extend_simps:
haftmann@43852:   "\<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))"
haftmann@43852:   "\<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))"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
haftmann@43852:   "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
haftmann@43817:   "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852:   "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852:   "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852:   "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852:   "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
haftmann@43852:   "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860:   by auto
paulson@13860: 
haftmann@43872: text {* Finally *}
haftmann@43872: 
haftmann@32135: no_notation
haftmann@46691:   less_eq (infix "\<sqsubseteq>" 50) and
haftmann@46691:   less (infix "\<sqsubset>" 50)
haftmann@32135: 
haftmann@30596: lemmas mem_simps =
haftmann@30596:   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596:   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596:   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669: 
wenzelm@11979: end
haftmann@49905: 




