
haftmann@32139: (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
wenzelm@11979: 
haftmann@32139: header {* Complete lattices, with special focus on sets *}
haftmann@32077: 
haftmann@32139: theory Complete_Lattice
haftmann@32139: imports Set
haftmann@32139: begin
haftmann@32077: 
haftmann@32077: notation
haftmann@34007:   less_eq (infix "\<sqsubseteq>" 50) and
haftmann@32077:   less (infix "\<sqsubset>" 50) and
haftmann@34007:   inf (infixl "\<sqinter>" 70) and
haftmann@34007:   sup (infixl "\<squnion>" 65) and
haftmann@32678:   top ("\<top>") and
haftmann@32678:   bot ("\<bottom>")
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@32139: subsection {* Abstract complete lattices *}
haftmann@32139: 
haftmann@34007: class complete_lattice = bounded_lattice + Inf + Sup +
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@32077: begin
haftmann@32077: 
haftmann@32678: lemma dual_complete_lattice:
haftmann@36635:   "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
haftmann@36635:   by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
haftmann@34007:     (unfold_locales, (fact bot_least top_greatest
haftmann@34007:         Sup_upper Sup_least Inf_lower Inf_greatest)+)
haftmann@32678: 
haftmann@34007: lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
haftmann@32077:   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@32077: 
haftmann@34007: lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
haftmann@32077:   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@32077: 
haftmann@41080: lemma Inf_empty [simp]:
haftmann@34007:   "\<Sqinter>{} = \<top>"
haftmann@34007:   by (auto intro: antisym Inf_greatest)
haftmann@32077: 
haftmann@41080: lemma Sup_empty [simp]:
haftmann@34007:   "\<Squnion>{} = \<bottom>"
haftmann@34007:   by (auto intro: antisym Sup_least)
haftmann@32077: 
haftmann@41080: lemma Inf_UNIV [simp]:
haftmann@41080:   "\<Sqinter>UNIV = \<bottom>"
haftmann@41080:   by (simp add: Sup_Inf Sup_empty [symmetric])
haftmann@41080: 
haftmann@41080: lemma Sup_UNIV [simp]:
haftmann@41080:   "\<Squnion>UNIV = \<top>"
haftmann@41080:   by (simp add: Inf_Sup Inf_empty [symmetric])
haftmann@41080: 
haftmann@32077: lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
haftmann@32077:   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
haftmann@32077: 
haftmann@32077: lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
haftmann@32077:   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
haftmann@32077: 
haftmann@32077: lemma Inf_singleton [simp]:
haftmann@32077:   "\<Sqinter>{a} = a"
haftmann@32077:   by (auto intro: antisym Inf_lower Inf_greatest)
haftmann@32077: 
haftmann@32077: lemma Sup_singleton [simp]:
haftmann@32077:   "\<Squnion>{a} = a"
haftmann@32077:   by (auto intro: antisym Sup_upper Sup_least)
haftmann@32077: 
haftmann@32077: lemma Inf_binary:
haftmann@32077:   "\<Sqinter>{a, b} = a \<sqinter> b"
haftmann@34007:   by (simp add: Inf_empty Inf_insert)
haftmann@32077: 
haftmann@32077: lemma Sup_binary:
haftmann@32077:   "\<Squnion>{a, b} = a \<squnion> b"
haftmann@34007:   by (simp add: Sup_empty Sup_insert)
haftmann@32077: 
haftmann@43754: lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
huffman@35629:   by (auto intro: Inf_greatest dest: Inf_lower)
huffman@35629: 
haftmann@43741: lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
haftmann@41082:   by (auto intro: Sup_least dest: Sup_upper)
hoelzl@38705: 
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@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: 
hoelzl@41971: lemma top_le:
haftmann@43741:   "\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>"
hoelzl@41971:   by (rule antisym) auto
hoelzl@41971: 
hoelzl@41971: lemma le_bot:
haftmann@43741:   "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
hoelzl@41971:   by (rule antisym) auto
hoelzl@41971: 
haftmann@43741: lemma not_less_bot[simp]: "\<not> (x \<sqsubset> \<bottom>)"
hoelzl@41971:   using bot_least[of x] by (auto simp: le_less)
hoelzl@41971: 
haftmann@43741: lemma not_top_less[simp]: "\<not> (\<top> \<sqsubset> x)"
hoelzl@41971:   using top_greatest[of x] by (auto simp: le_less)
hoelzl@41971: 
haftmann@43741: lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
hoelzl@41971:   using Sup_upper[of u A] by auto
hoelzl@41971: 
haftmann@43741: lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
hoelzl@41971:   using Inf_lower[of u A] by auto
hoelzl@41971: 
haftmann@32077: definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@32117:   "INFI A f = \<Sqinter> (f ` A)"
haftmann@32077: 
haftmann@41082: definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@41082:   "SUPR A f = \<Squnion> (f ` A)"
haftmann@41082: 
haftmann@32077: end
haftmann@32077: 
haftmann@32077: syntax
haftmann@41082:   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@41082:   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
haftmann@41080:   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@41080:   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
haftmann@41080: 
haftmann@41080: syntax (xsymbols)
haftmann@41082:   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082:   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080:   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080:   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077: 
haftmann@32077: translations
haftmann@41082:   "INF x y. B"   == "INF x. INF y. B"
haftmann@41082:   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
haftmann@41082:   "INF x. B"     == "INF x:CONST UNIV. B"
haftmann@41082:   "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@32077:   "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@32077:   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
haftmann@32077:   "SUP x. B"     == "SUP x:CONST UNIV. B"
haftmann@32077:   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@32077: 
wenzelm@35115: print_translation {*
wenzelm@42284:   [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
wenzelm@42284:     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
wenzelm@35115: *} -- {* to avoid eta-contraction of body *}
wenzelm@11979: 
haftmann@32077: context complete_lattice
haftmann@32077: begin
haftmann@32077: 
hoelzl@41971: lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
hoelzl@41971:   by (simp add: SUPR_def cong: image_cong)
hoelzl@41971: 
hoelzl@41971: lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
hoelzl@41971:   by (simp add: INFI_def cong: image_cong)
hoelzl@41971: 
haftmann@43741: lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
haftmann@32077:   by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@32077: 
haftmann@43741: lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
hoelzl@41971:   using le_SUPI[of i A M] by auto
hoelzl@41971: 
haftmann@43741: lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"
haftmann@32077:   by (auto simp add: SUPR_def intro: Sup_least)
haftmann@32077: 
haftmann@43741: lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"
haftmann@32077:   by (auto simp add: INFI_def intro: Inf_lower)
haftmann@32077: 
haftmann@43741: lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"
hoelzl@41971:   using INF_leI[of i A M] by auto
hoelzl@41971: 
haftmann@43741: lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"
haftmann@32077:   by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@32077: 
haftmann@43753: lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
huffman@35629:   unfolding SUPR_def by (auto simp add: Sup_le_iff)
huffman@35629: 
haftmann@43753: lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
huffman@35629:   unfolding INFI_def by (auto simp add: le_Inf_iff)
huffman@35629: 
haftmann@43753: lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"
haftmann@32077:   by (auto intro: antisym INF_leI le_INFI)
haftmann@32077: 
haftmann@43753: lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"
haftmann@41082:   by (auto intro: antisym SUP_leI le_SUPI)
hoelzl@38705: 
hoelzl@38705: lemma INF_mono:
haftmann@43753:   "(\<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)"
hoelzl@38705:   by (force intro!: Inf_mono simp: INFI_def)
hoelzl@38705: 
haftmann@41082: lemma SUP_mono:
haftmann@43753:   "(\<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)"
haftmann@41082:   by (force intro!: Sup_mono simp: SUPR_def)
hoelzl@40872: 
haftmann@43753: lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
hoelzl@40872:   by (intro INF_mono) auto
hoelzl@40872: 
haftmann@43753: lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
haftmann@41082:   by (intro SUP_mono) auto
hoelzl@40872: 
haftmann@43753: 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)"
hoelzl@40872:   by (iprover intro: INF_leI le_INFI order_trans antisym)
hoelzl@40872: 
haftmann@43753: 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@41082:   by (iprover intro: SUP_leI le_SUPI order_trans antisym)
haftmann@41082: 
haftmann@32077: end
haftmann@32077: 
haftmann@41082: lemma Inf_less_iff:
haftmann@41082:   fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753:   shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43754:   unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@41082: 
hoelzl@38705: lemma less_Sup_iff:
hoelzl@38705:   fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753:   shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43754:   unfolding not_le [symmetric] Sup_le_iff by auto
hoelzl@38705: 
haftmann@41082: lemma INF_less_iff:
haftmann@41082:   fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753:   shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@41082:   unfolding INFI_def Inf_less_iff by auto
haftmann@32077: 
hoelzl@40872: lemma less_SUP_iff:
hoelzl@40872:   fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753:   shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
hoelzl@40872:   unfolding SUPR_def less_Sup_iff by auto
hoelzl@40872: 
haftmann@32139: subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077: 
haftmann@32077: instantiation bool :: complete_lattice
haftmann@32077: begin
haftmann@32077: 
haftmann@32077: definition
haftmann@41080:   "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077: 
haftmann@32077: definition
haftmann@41080:   "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077: 
haftmann@32077: instance proof
haftmann@32077: qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@32077: 
haftmann@32077: end
haftmann@32077: 
haftmann@41080: lemma INFI_bool_eq [simp]:
haftmann@32120:   "INFI = Ball"
haftmann@32120: proof (rule ext)+
haftmann@32120:   fix A :: "'a set"
haftmann@32120:   fix P :: "'a \<Rightarrow> bool"
haftmann@43753:   show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@32120:     by (auto simp add: Ball_def INFI_def Inf_bool_def)
haftmann@32120: qed
haftmann@32120: 
haftmann@41080: lemma SUPR_bool_eq [simp]:
haftmann@32120:   "SUPR = Bex"
haftmann@32120: proof (rule ext)+
haftmann@32120:   fix A :: "'a set"
haftmann@32120:   fix P :: "'a \<Rightarrow> bool"
haftmann@43753:   show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@32120:     by (auto simp add: Bex_def SUPR_def Sup_bool_def)
haftmann@32120: qed
haftmann@32120: 
haftmann@32077: instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077: begin
haftmann@32077: 
haftmann@32077: definition
haftmann@41080:   "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080: 
haftmann@41080: lemma Inf_apply:
haftmann@41080:   "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080:   by (simp add: Inf_fun_def)
haftmann@32077: 
haftmann@32077: definition
haftmann@41080:   "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080: 
haftmann@41080: lemma Sup_apply:
haftmann@41080:   "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080:   by (simp add: Sup_fun_def)
haftmann@32077: 
haftmann@32077: instance proof
haftmann@41080: qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077:   intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077: 
haftmann@32077: end
haftmann@32077: 
haftmann@41080: lemma INFI_apply:
haftmann@41080:   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@41080:   by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
hoelzl@38705: 
haftmann@41080: lemma SUPR_apply:
haftmann@41080:   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@41080:   by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
haftmann@32077: 
haftmann@32077: 
haftmann@41082: subsection {* 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@41082:   Inter  ("\<Inter>_" [90] 90)
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@41082:     by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082: qed
haftmann@41082: 
haftmann@43741: lemma Inter_iff [simp,no_atp]: "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@43740: lemma (in complete_lattice) Inf_less_eq:
haftmann@43740:   assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43740:     and "A \<noteq> {}"
haftmann@43753:   shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43740: proof -
haftmann@43740:   from `A \<noteq> {}` obtain v where "v \<in> A" by blast
haftmann@43740:   moreover with assms have "v \<sqsubseteq> u" by blast
haftmann@43740:   ultimately show ?thesis by (rule Inf_lower2)
haftmann@43740: qed
haftmann@41082: 
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: 
haftmann@41082: lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739:   by (fact Inf_binary [symmetric])
haftmann@41082: 
haftmann@41082: lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082:   by (fact Inf_empty)
haftmann@41082: 
haftmann@41082: lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739:   by (fact Inf_UNIV)
haftmann@41082: 
haftmann@41082: lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739:   by (fact Inf_insert)
haftmann@41082: 
haftmann@43741: lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43741:   by (auto intro: Inf_greatest Inf_lower)
haftmann@43741: 
haftmann@41082: lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43741:   by (fact Inf_inter_less)
haftmann@43741: 
haftmann@43756: lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43756:   by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
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: 
haftmann@43801: lemma (in complete_lattice) Inf_top_conv [no_atp]:
haftmann@43801:   "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801:   "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801: proof -
haftmann@43801:   show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801:   proof
haftmann@43801:     assume "\<forall>x\<in>A. x = \<top>"
haftmann@43801:     then have "A = {} \<or> A = {\<top>}" by auto
haftmann@43801:     then show "\<Sqinter>A = \<top>" by auto
haftmann@43801:   next
haftmann@43801:     assume "\<Sqinter>A = \<top>"
haftmann@43801:     show "\<forall>x\<in>A. x = \<top>"
haftmann@43801:     proof (rule ccontr)
haftmann@43801:       assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801:       then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43801:       then obtain B where "A = insert x B" by blast
haftmann@43801:       with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
haftmann@43801:     qed
haftmann@43801:   qed
haftmann@43801:   then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43801: qed
haftmann@41082: 
haftmann@41082: lemma Inter_UNIV_conv [simp,no_atp]:
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@43756: lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
haftmann@43756:   by (auto intro: Inf_greatest Inf_lower)
haftmann@43756: 
haftmann@43741: lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43756:   by (fact Inf_anti_mono)
haftmann@41082: 
haftmann@41082: 
haftmann@41082: subsection {* 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@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@41082: lemma INTER_eq_Inter_image:
haftmann@41082:   "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@41082:   by (fact INFI_def)
haftmann@41082:   
haftmann@41082: lemma Inter_def:
haftmann@41082:   "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082:   by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082: 
haftmann@41082: lemma INTER_def:
haftmann@41082:   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082:   by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082: 
haftmann@41082: lemma Inter_image_eq [simp]:
haftmann@41082:   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43801:   by (rule sym) (fact INFI_def)
haftmann@41082: 
haftmann@41082: lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
haftmann@41082:   by (unfold INTER_def) blast
haftmann@41082: 
haftmann@41082: lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
haftmann@41082:   by (unfold INTER_def) blast
haftmann@41082: 
haftmann@41082: lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
haftmann@41082:   by auto
haftmann@41082: 
haftmann@41082: lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
haftmann@41082:   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
haftmann@41082:   by (unfold INTER_def) blast
haftmann@41082: 
haftmann@41082: lemma INT_cong [cong]:
haftmann@41082:     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
haftmann@41082:   by (simp add: INTER_def)
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@41082: lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082:   by (fact INF_leI)
haftmann@41082: 
haftmann@41082: lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@41082:   by (fact le_INFI)
haftmann@41082: 
haftmann@41082: lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<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@41082:   by blast
haftmann@41082: 
haftmann@41082: 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@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma INT_insert_distrib:
haftmann@41082:     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@41082:   by auto
haftmann@41082: 
haftmann@41082: lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082:   -- {* Look: it has an \emph{existential} quantifier *}
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: lemma INTER_UNIV_conv[simp]:
haftmann@41082:  "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082:  "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082: by blast+
haftmann@41082: 
haftmann@41082: lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@41082:   by (auto intro: bool_induct)
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@41082: lemma INT_anti_mono:
haftmann@41082:   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@41082:     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@41082:   -- {* The last inclusion is POSITIVE! *}
haftmann@41082:   by (blast dest: subsetD)
haftmann@41082: 
haftmann@41082: lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@41082:   by blast
haftmann@41082: 
haftmann@41082: 
haftmann@32139: subsection {* 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@32115:   Union  ("\<Union>_" [90] 90)
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@32587:     by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115: qed
haftmann@32115: 
blanchet@35828: lemma Union_iff [simp, no_atp]:
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@32115:   "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@32135: lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
haftmann@32135:   by (iprover intro: subsetI UnionI)
haftmann@32135: 
haftmann@32135: lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
haftmann@32135:   by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135: 
haftmann@32135: lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_empty [simp]: "Union({}) = {}"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_UNIV [simp]: "Union UNIV = UNIV"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135:   by blast
haftmann@32135: 
blanchet@35828: lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
haftmann@32135:   by blast
haftmann@32135: 
blanchet@35828: lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135:   by blast
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@32135: lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32115: 
haftmann@32139: subsection {* 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@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
haftmann@32077:   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077:   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>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: 
haftmann@32135: lemma UNION_eq_Union_image:
haftmann@32135:   "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32606:   by (fact SUPR_def)
haftmann@32115: 
haftmann@32115: lemma Union_def:
haftmann@32117:   "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115:   by (simp add: UNION_eq_Union_image image_def)
haftmann@32115: 
blanchet@35828: lemma UNION_def [no_atp]:
haftmann@32135:   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117:   by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115:   
haftmann@32115: lemma Union_image_eq [simp]:
haftmann@32115:   "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115:   by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115:   
wenzelm@11979: lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979:   by (unfold UNION_def) blast
wenzelm@11979: 
wenzelm@11979: lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x: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: 
wenzelm@11979: lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979:   by (unfold UNION_def) blast
clasohm@923: 
wenzelm@11979: lemma UN_cong [cong]:
wenzelm@11979:     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979:   by (simp add: UNION_def)
wenzelm@11979: 
berghofe@29691: lemma strong_UN_cong:
berghofe@29691:     "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691:   by (simp add: UNION_def simp_implies_def)
berghofe@29691: 
haftmann@32077: lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077:   by blast
haftmann@32077: 
haftmann@32135: lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606:   by (fact le_SUPI)
haftmann@32135: 
haftmann@32135: lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135:   by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135: 
blanchet@35828: lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135:   by blast
haftmann@32135: 
blanchet@35828: lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135:   by auto
haftmann@32135: 
haftmann@32135: lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: 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@32135:   by blast
haftmann@32135: 
haftmann@32135: 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 image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135:   by auto
haftmann@32135: 
haftmann@32135: lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma UNION_empty_conv[simp]:
haftmann@32135:   "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135:   "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135: by blast+
haftmann@32135: 
blanchet@35828: lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<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@32135: lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135:   by (auto intro: bool_contrapos)
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@32135:   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135:     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135:   by (blast dest: subsetD)
haftmann@32135: 
haftmann@32135: lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135:   -- {* NOT suitable for rewriting *}
haftmann@32135:   by blast
haftmann@32135: 
haftmann@32135: lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135: by blast
haftmann@32135: 
wenzelm@11979: 
haftmann@32139: subsection {* Distributive laws *}
wenzelm@12897: 
wenzelm@12897: lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 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 *}
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 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)"
wenzelm@12897:   -- {* Equivalent version *}
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 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)"
wenzelm@12897:   -- {* Equivalent version *}
wenzelm@12897:   by blast
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. *}
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897:   by blast
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)"
wenzelm@12897:   by blast
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)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 
haftmann@32139: subsection {* Complement *}
haftmann@32135: 
haftmann@32135: lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897:   by blast
wenzelm@12897: 
haftmann@32135: lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 
haftmann@32139: subsection {* 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]:
wenzelm@12897:   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897:   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897:   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897:   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897:   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897:   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897:   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897:   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897:   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897:   "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897:   by auto
wenzelm@12897: 
wenzelm@12897: lemma INT_simps [simp]:
wenzelm@12897:   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897:   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897:   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897:   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897:   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897:   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897:   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897:   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897:   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897:   "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897:   by auto
wenzelm@12897: 
blanchet@35828: lemma ball_simps [simp,no_atp]:
wenzelm@12897:   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897:   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897:   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897:   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897:   "!!P. (ALL x:{}. P x) = True"
wenzelm@12897:   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897:   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897:   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897:   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897:   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897:   "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897:   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897:   by auto
wenzelm@12897: 
blanchet@35828: lemma bex_simps [simp,no_atp]:
wenzelm@12897:   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897:   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897:   "!!P. (EX x:{}. P x) = False"
wenzelm@12897:   "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897:   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897:   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897:   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897:   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897:   "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897:   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897:   by auto
wenzelm@12897: 
wenzelm@12897: lemma ball_conj_distrib:
wenzelm@12897:   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: lemma bex_disj_distrib:
wenzelm@12897:   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897:   by blast
wenzelm@12897: 
wenzelm@12897: 
paulson@13860: text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860: 
paulson@13860: lemma UN_extend_simps:
paulson@13860:   "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860:   "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860:   "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860:   "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860:   "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860:   "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860:   "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860:   "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860:   "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860:   "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860:   by auto
paulson@13860: 
paulson@13860: lemma INT_extend_simps:
paulson@13860:   "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860:   "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860:   "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860:   "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860:   "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860:   "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860:   "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860:   "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860:   "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860:   "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860:   by auto
paulson@13860: 
paulson@13860: 
haftmann@32135: no_notation
haftmann@32135:   less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135:   less (infix "\<sqsubset>" 50) and
haftmann@41082:   bot ("\<bottom>") and
haftmann@41082:   top ("\<top>") and
haftmann@32135:   inf  (infixl "\<sqinter>" 70) and
haftmann@32135:   sup  (infixl "\<squnion>" 65) and
haftmann@32135:   Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082:   Sup  ("\<Squnion>_" [900] 900)
haftmann@32135: 
haftmann@41080: no_syntax (xsymbols)
haftmann@41082:   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082:   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080:   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080:   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080: 
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




