src/HOL/Complete_Lattice.thy
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
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header {* Complete lattices, with special focus on sets *}
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theory Complete_Lattice
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imports Set
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begin
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notation
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  less_eq (infix "\<sqsubseteq>" 50) and
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  less (infix "\<sqsubset>" 50) and
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  inf (infixl "\<sqinter>" 70) and
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  sup (infixl "\<squnion>" 65) and
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  top ("\<top>") and
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  bot ("\<bottom>")
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subsection {* Syntactic infimum and supremum operations *}
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class Inf =
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  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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class Sup =
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  fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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subsection {* Abstract complete lattices *}
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class complete_lattice = bounded_lattice + Inf + Sup +
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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begin
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lemma dual_complete_lattice:
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  "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
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  by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
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    (unfold_locales, (fact bot_least top_greatest
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        Sup_upper Sup_least Inf_lower Inf_greatest)+)
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
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  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_empty [simp]:
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  "\<Sqinter>{} = \<top>"
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  by (auto intro: antisym Inf_greatest)
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lemma Sup_empty [simp]:
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  "\<Squnion>{} = \<bottom>"
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  by (auto intro: antisym Sup_least)
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lemma Inf_UNIV [simp]:
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  "\<Sqinter>UNIV = \<bottom>"
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  by (simp add: Sup_Inf Sup_empty [symmetric])
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lemma Sup_UNIV [simp]:
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  "\<Squnion>UNIV = \<top>"
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  by (simp add: Inf_Sup Inf_empty [symmetric])
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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
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lemma Inf_singleton [simp]:
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  "\<Sqinter>{a} = a"
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  by (auto intro: antisym Inf_lower Inf_greatest)
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lemma Sup_singleton [simp]:
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  "\<Squnion>{a} = a"
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  by (auto intro: antisym Sup_upper Sup_least)
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lemma Inf_binary:
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  "\<Sqinter>{a, b} = a \<sqinter> b"
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  by (simp add: Inf_insert)
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lemma Sup_binary:
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  "\<Squnion>{a, b} = a \<squnion> b"
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  by (simp add: Sup_insert)
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
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  by (auto intro: Inf_greatest dest: Inf_lower)
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lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
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  by (auto intro: Sup_least dest: Sup_upper)
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lemma Inf_mono:
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
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  shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
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proof (rule Inf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast
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  from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower)
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  with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto
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qed
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lemma Sup_mono:
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  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
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  shows "\<Squnion>A \<sqsubseteq> \<Squnion>B"
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proof (rule Sup_least)
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  fix a assume "a \<in> A"
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  with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast
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  from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper)
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  with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto
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qed
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lemma top_le:
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  "\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>"
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  by (rule antisym) auto
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lemma le_bot:
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  "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
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  by (rule antisym) auto
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lemma not_less_bot [simp]: "\<not> (x \<sqsubset> \<bottom>)"
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  using bot_least [of x] by (auto simp: le_less)
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lemma not_top_less [simp]: "\<not> (\<top> \<sqsubset> x)"
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  using top_greatest [of x] by (auto simp: le_less)
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
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  using Sup_upper[of u A] by auto
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
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  using Inf_lower[of u A] by auto
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  "INFI A f = \<Sqinter> (f ` A)"
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  "SUPR A f = \<Squnion> (f ` A)"
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end
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syntax
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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translations
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  "INF x y. B"   == "INF x. INF y. B"
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  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
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  "INF x. B"     == "INF x:CONST UNIV. B"
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  "INF x:A. B"   == "CONST INFI A (%x. B)"
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  "SUP x y. B"   == "SUP x. SUP y. B"
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  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
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  "SUP x. B"     == "SUP x:CONST UNIV. B"
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  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
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print_translation {*
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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*} -- {* to avoid eta-contraction of body *}
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context complete_lattice
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begin
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lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
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  by (simp add: SUPR_def cong: image_cong)
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lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
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  by (simp add: INFI_def cong: image_cong)
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lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
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  by (auto simp add: SUPR_def intro: Sup_upper)
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lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
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  using le_SUPI[of i A M] by auto
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lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"
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  by (auto simp add: SUPR_def intro: Sup_least)
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"
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  by (auto simp add: INFI_def intro: Inf_lower)
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lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"
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  using INF_leI[of i A M] by auto
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lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"
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  by (auto simp add: INFI_def intro: Inf_greatest)
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
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  unfolding SUPR_def by (auto simp add: Sup_le_iff)
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
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  unfolding INFI_def by (auto simp add: le_Inf_iff)
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lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"
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  by (auto intro: antisym INF_leI le_INFI)
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lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"
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  by (auto intro: antisym SUP_leI le_SUPI)
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lemma INF_mono:
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  "(\<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)"
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  by (force intro!: Inf_mono simp: INFI_def)
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lemma SUP_mono:
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  "(\<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)"
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  by (force intro!: Sup_mono simp: SUPR_def)
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lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
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  by (intro INF_mono) auto
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lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
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  by (intro SUP_mono) auto
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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)"
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  by (iprover intro: INF_leI le_INFI order_trans antisym)
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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)"
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  by (iprover intro: SUP_leI le_SUPI order_trans antisym)
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end
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lemma Inf_less_iff:
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  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
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  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
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  unfolding not_le [symmetric] le_Inf_iff by auto
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lemma less_Sup_iff:
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  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
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  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
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  unfolding not_le [symmetric] Sup_le_iff by auto
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lemma INF_less_iff:
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  fixes a :: "'a::{complete_lattice,linorder}"
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  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
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  unfolding INFI_def Inf_less_iff by auto
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lemma less_SUP_iff:
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  fixes a :: "'a::{complete_lattice,linorder}"
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  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
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  unfolding SUPR_def less_Sup_iff by auto
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subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
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instantiation bool :: complete_lattice
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begin
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definition
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  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
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definition
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  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
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instance proof
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qed (auto simp add: Inf_bool_def Sup_bool_def)
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end
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lemma INFI_bool_eq [simp]:
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  "INFI = Ball"
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proof (rule ext)+
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  fix A :: "'a set"
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  fix P :: "'a \<Rightarrow> bool"
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  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
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    by (auto simp add: Ball_def INFI_def Inf_bool_def)
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qed
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lemma SUPR_bool_eq [simp]:
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  "SUPR = Bex"
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proof (rule ext)+
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  fix A :: "'a set"
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  fix P :: "'a \<Rightarrow> bool"
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  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
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    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
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qed
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instantiation "fun" :: (type, complete_lattice) complete_lattice
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begin
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definition
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  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
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lemma Inf_apply:
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  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
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  by (simp add: Inf_fun_def)
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definition
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  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
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lemma Sup_apply:
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  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
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  by (simp add: Sup_fun_def)
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instance proof
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qed (auto simp add: le_fun_def Inf_apply Sup_apply
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  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
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end
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lemma INFI_apply:
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  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
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  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
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lemma SUPR_apply:
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  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
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  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
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subsection {* Inter *}
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abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
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  "Inter S \<equiv> \<Sqinter>S"
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notation (xsymbols)
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  Inter  ("\<Inter>_" [90] 90)
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lemma Inter_eq:
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  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
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proof (rule set_eqI)
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   324
  fix x
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  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
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   326
    by auto
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  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
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    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
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qed
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   330
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lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
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  by (unfold Inter_eq) blast
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lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
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  by (simp add: Inter_eq)
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text {*
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  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
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  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
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  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
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*}
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   342
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lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
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  by auto
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lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
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  -- {* ``Classical'' elimination rule -- does not require proving
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    @{prop "X \<in> C"}. *}
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  by (unfold Inter_eq) blast
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lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
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  by (fact Inf_lower)
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   353
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lemma (in complete_lattice) Inf_less_eq:
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  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
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   356
    and "A \<noteq> {}"
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   357
  shows "\<Sqinter>A \<sqsubseteq> u"
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   358
proof -
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   359
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
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   360
  moreover with assms have "v \<sqsubseteq> u" by blast
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   361
  ultimately show ?thesis by (rule Inf_lower2)
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qed
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   363
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   364
lemma Inter_subset:
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   365
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
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   366
  by (fact Inf_less_eq)
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   367
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lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
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   369
  by (fact Inf_greatest)
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   370
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lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
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   372
  by (fact Inf_binary [symmetric])
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   373
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lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
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  by (fact Inf_empty)
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   376
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   377
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
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   378
  by (fact Inf_UNIV)
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   379
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   380
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
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   381
  by (fact Inf_insert)
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   382
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   383
lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
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diff changeset
   384
  by (auto intro: Inf_greatest Inf_lower)
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diff changeset
   385
41082
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   386
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
43741
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diff changeset
   387
  by (fact Inf_inter_less)
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diff changeset
   388
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   389
lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
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diff changeset
   390
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
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   391
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   392
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
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diff changeset
   393
  by (fact Inf_union_distrib)
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diff changeset
   394
43801
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   395
lemma (in complete_lattice) Inf_top_conv [no_atp]:
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diff changeset
   396
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
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diff changeset
   397
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
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diff changeset
   398
proof -
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diff changeset
   399
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
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haftmann
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diff changeset
   400
  proof
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diff changeset
   401
    assume "\<forall>x\<in>A. x = \<top>"
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   402
    then have "A = {} \<or> A = {\<top>}" by auto
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haftmann
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diff changeset
   403
    then show "\<Sqinter>A = \<top>" by auto
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haftmann
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diff changeset
   404
  next
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diff changeset
   405
    assume "\<Sqinter>A = \<top>"
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haftmann
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diff changeset
   406
    show "\<forall>x\<in>A. x = \<top>"
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haftmann
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diff changeset
   407
    proof (rule ccontr)
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haftmann
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diff changeset
   408
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   409
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   410
      then obtain B where "A = insert x B" by blast
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   411
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
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diff changeset
   412
    qed
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diff changeset
   413
  qed
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   414
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
097732301fc0 more generalization towards complete lattices
haftmann
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diff changeset
   415
qed
41082
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diff changeset
   416
9ff94e7cc3b3 bot comes before top, inf before sup etc.
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diff changeset
   417
lemma Inter_UNIV_conv [simp,no_atp]:
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   418
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   419
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
43801
097732301fc0 more generalization towards complete lattices
haftmann
parents: 43756
diff changeset
   420
  by (fact Inf_top_conv)+
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   421
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   422
lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   423
  by (auto intro: Inf_greatest Inf_lower)
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   424
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   425
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   426
  by (fact Inf_anti_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   427
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   428
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   429
subsection {* Intersections of families *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   430
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   431
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   432
  "INTER \<equiv> INFI"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   433
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   434
syntax
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   435
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   436
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   437
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   438
syntax (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   439
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   440
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   441
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   442
syntax (latex output)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   443
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   444
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   445
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   446
translations
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   447
  "INT x y. B"  == "INT x. INT y. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   448
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   449
  "INT x. B"    == "INT x:CONST UNIV. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   450
  "INT x:A. B"  == "CONST INTER A (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   451
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   452
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   453
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   454
*} -- {* to avoid eta-contraction of body *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   455
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   456
lemma INTER_eq_Inter_image:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   457
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   458
  by (fact INFI_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   459
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   460
lemma Inter_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   461
  "\<Inter>S = (\<Inter>x\<in>S. x)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   462
  by (simp add: INTER_eq_Inter_image image_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   463
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   464
lemma INTER_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   465
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   466
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   467
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   468
lemma Inter_image_eq [simp]:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   469
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
43801
097732301fc0 more generalization towards complete lattices
haftmann
parents: 43756
diff changeset
   470
  by (rule sym) (fact INFI_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   471
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   472
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   473
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   474
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   475
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   476
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   477
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   478
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   479
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   480
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   481
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"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   482
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   483
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   484
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   485
lemma INT_cong [cong]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   486
    "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)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   487
  by (simp add: INTER_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   488
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   489
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   490
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   491
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   492
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   493
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   494
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   495
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   496
  by (fact INF_leI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   497
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   498
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   499
  by (fact le_INFI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   500
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   501
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   502
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   503
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   504
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   505
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   506
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   507
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   508
  by (fact le_INF_iff)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   509
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   510
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   511
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   512
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   513
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   514
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   515
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   516
lemma INT_insert_distrib:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   517
    "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   518
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   519
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   520
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   521
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   522
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   523
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   524
  -- {* Look: it has an \emph{existential} quantifier *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   525
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   526
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   527
lemma INTER_UNIV_conv[simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   528
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   529
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   530
by blast+
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   531
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   532
lemma INT_bool_eq: "(\<Inter>b. A b) = (A True \<inter> A False)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   533
  by (auto intro: bool_induct)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   534
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   535
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   536
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   537
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   538
lemma INT_anti_mono:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   539
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   540
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   541
  -- {* The last inclusion is POSITIVE! *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   542
  by (blast dest: subsetD)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   543
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   544
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   545
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   546
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   547
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   548
subsection {* Union *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   549
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   550
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   551
  "Union S \<equiv> \<Squnion>S"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   552
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   553
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   554
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   555
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   556
lemma Union_eq:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   557
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 38705
diff changeset
   558
proof (rule set_eqI)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   559
  fix x
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   560
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   561
    by auto
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   562
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   563
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   564
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   565
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   566
lemma Union_iff [simp, no_atp]:
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   567
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   568
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   569
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   570
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   571
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   572
  -- {* The order of the premises presupposes that @{term C} is rigid;
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   573
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   574
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   575
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   576
lemma UnionE [elim!]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   577
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   578
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   579
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   580
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   581
  by (iprover intro: subsetI UnionI)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   582
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   583
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   584
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   585
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   586
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   587
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   588
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   589
lemma Union_empty [simp]: "\<Union>{} = {}"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   590
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   591
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   592
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   593
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   594
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   595
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   596
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   597
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   598
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   599
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   600
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   601
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   602
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   603
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   604
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   605
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   606
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   607
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   608
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   609
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   610
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   611
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   612
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   613
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   614
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   615
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   616
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   617
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   618
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   619
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   620
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   621
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   622
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   623
subsection {* Unions of families *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   624
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   625
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   626
  "UNION \<equiv> SUPR"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   627
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   628
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   629
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   630
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   631
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   632
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   633
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   634
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   635
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   636
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   637
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   638
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   639
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   640
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   641
  "UN x y. B"   == "UN x. UN y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   642
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   643
  "UN x. B"     == "UN x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   644
  "UN x:A. B"   == "CONST UNION A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   645
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   646
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   647
  Note the difference between ordinary xsymbol syntax of indexed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   648
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   649
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   650
  former does not make the index expression a subscript of the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   651
  union/intersection symbol because this leads to problems with nested
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   652
  subscripts in Proof General.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   653
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   654
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   655
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   656
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   657
*} -- {* to avoid eta-contraction of body *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   658
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   659
lemma UNION_eq_Union_image:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   660
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   661
  by (fact SUPR_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   662
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   663
lemma Union_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   664
  "\<Union>S = (\<Union>x\<in>S. x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   665
  by (simp add: UNION_eq_Union_image image_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   666
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   667
lemma UNION_def [no_atp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   668
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   669
  by (auto simp add: UNION_eq_Union_image Union_eq)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   670
  
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   671
lemma Union_image_eq [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   672
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   673
  by (rule sym) (fact UNION_eq_Union_image)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   674
  
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   675
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   678
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   683
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"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   685
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   686
lemma UN_cong [cong]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   687
    "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)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   689
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   690
lemma strong_UN_cong:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   691
    "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)"
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   692
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   693
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   694
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   695
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   696
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   697
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   698
  by (fact le_SUPI)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   699
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   700
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   701
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   702
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   703
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   704
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   705
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   706
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   707
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   708
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   709
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   710
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   711
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   712
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   713
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   714
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   715
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   716
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   717
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   718
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   719
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   720
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   721
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   722
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   723
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   724
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)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   725
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   726
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   727
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)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   728
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   729
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   730
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   731
  by (fact SUP_le_iff)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   732
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   733
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   734
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   735
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   736
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   737
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   738
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   739
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   740
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   741
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   742
lemma UNION_empty_conv[simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   743
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   744
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   745
by blast+
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   746
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   747
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   748
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   749
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   750
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   751
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   752
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   753
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   754
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   755
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   756
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   757
  by (auto simp add: split_if_mem2)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   758
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   759
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   760
  by (auto intro: bool_contrapos)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   761
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   762
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   763
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   764
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   765
lemma UN_mono:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   766
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   767
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   768
  by (blast dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   769
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   770
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   771
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   772
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   773
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   774
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   775
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   776
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   777
  -- {* NOT suitable for rewriting *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   778
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   779
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   780
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   781
  by blast
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   782
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   784
subsection {* Distributive laws *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   785
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   786
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   787
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   788
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   789
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   790
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   791
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   792
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   793
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   794
  -- {* Union of a family of unions *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   795
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   796
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   797
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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   798
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   799
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   800
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   801
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   802
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   803
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   804
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   805
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   806
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   807
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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   808
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   809
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   810
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   811
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   812
  -- {* Halmos, Naive Set Theory, page 35. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   813
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   814
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   815
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   816
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   817
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   818
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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   819
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   820
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   821
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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   822
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   823
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   824
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   825
subsection {* Complement *}
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   826
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   827
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   828
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   829
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   830
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   831
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   832
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   833
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   834
subsection {* Miniscoping and maxiscoping *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   835
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   836
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   837
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   838
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   839
lemma UN_simps [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   840
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   841
  "\<And>A B C. (\<Union>x\<in>C. A x \<union>  B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   842
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   843
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   844
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   845
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   846
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   847
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   848
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
   849
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   850
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   851
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   852
lemma INT_simps [simp]:
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
   853
  "\<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)"
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
   854
  "\<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))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   855
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   856
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   857
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   858
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   859
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   860
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   861
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   862
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   863
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   864
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   865
lemma ball_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   866
  "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   867
  "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   868
  "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   869
  "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   870
  "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   871
  "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   872
  "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   873
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   874
  "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   875
  "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   876
  "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   877
  "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   878
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   879
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   880
lemma bex_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   881
  "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   882
  "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   883
  "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   884
  "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   885
  "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   886
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   887
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   888
  "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   889
  "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   890
  "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   891
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   892
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   893
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   894
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   895
lemma UN_extend_simps:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   896
  "\<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)))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   897
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   898
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   899
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   900
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   901
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   902
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   903
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   904
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
   905
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   906
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   907
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   908
lemma INT_extend_simps:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   909
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   910
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   911
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   912
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   913
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   914
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   915
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   916
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   917
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   918
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   919
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   920
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   921
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   922
no_notation
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   923
  less_eq  (infix "\<sqsubseteq>" 50) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   924
  less (infix "\<sqsubset>" 50) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   925
  bot ("\<bottom>") and
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   926
  top ("\<top>") and
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   927
  inf  (infixl "\<sqinter>" 70) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   928
  sup  (infixl "\<squnion>" 65) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   929
  Inf  ("\<Sqinter>_" [900] 900) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   930
  Sup  ("\<Squnion>_" [900] 900)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   931
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   932
no_syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   933
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   934
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   935
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   936
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   937
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   938
lemmas mem_simps =
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   939
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   940
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   941
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
   942
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   943
end